Prime Factorization Calculator

Please provide an integer to find its prime factors as well as a factor tree.

Related Factor Calculator | Common Factor Calculator

What is a prime number?

Prime numbers are natural numbers (positive whole numbers that sometimes include 0 in certain definitions) that are greater than 1, that cannot be formed by multiplying two smaller numbers. An example of a prime number is 7, since it can only be formed by multiplying the numbers 1 and 7. Other examples include 2, 3, 5, 11, etc.

Numbers that can be formed with two other natural numbers, that are greater than 1, are called composite numbers. Examples of this include numbers like, 4, 6, 9, etc.

Prime numbers are widely used in number theory due to the fundamental theorem of arithmetic. This theorem states that natural numbers greater than 1 are either prime, or can be factored as a product of prime numbers. As an example, the number 60 can be factored into a product of prime numbers as follows:

60 = 5 × 3 × 2 × 2

As can be seen from the example above, there are no composite numbers in the factorization.

What is prime factorization?

Prime factorization is the decomposition of a composite number into a product of prime numbers. There are many factoring algorithms, some more complicated than others.

Trial division:

One method for finding the prime factors of a composite number is trial division. Trial division is one of the more basic algorithms, though it is highly tedious. It involves testing each integer by dividing the composite number in question by the integer, and determining if, and how many times, the integer can divide the number evenly. As a simple example, below is the prime factorization of 820 using trial division:

820 ÷ 2 = 410

410 ÷ 2 = 205

Since 205 is no longer divisible by 2, test the next integers. 205 cannot be evenly divided by 3. 4 is not a prime number. It can however be divided by 5:

205 ÷ 5 = 41

Since 41 is a prime number, this concludes the trial division. Thus:

820 = 41 × 5 × 2 × 2

The products can also be written as:

820 = 41 × 5 × 2 2

This is essentially the "brute force" method for determining the prime factors of a number, and though 820 is a simple example, it can get far more tedious very quickly.

Prime decomposition:

Another common way to conduct prime factorization is referred to as prime decomposition, and can involve the use of a factor tree. Creating a factor tree involves breaking up the composite number into factors of the composite number, until all of the numbers are prime. In the example below, the prime factors are found by dividing 820 by a prime factor, 2, then continuing to divide the result until all factors are prime. The example below demonstrates two ways that a factor tree can be created using the number 820:

prime factorization of 820

Thus, it can be seen that the prime factorization of 820, in either case, again is:

While these methods work for smaller numbers (and there are many other algorithms), there is no known algorithm for much larger numbers, and it can take a long period of time for even machines to compute the prime factorizations of larger numbers; in 2009, scientists concluded a project using hundreds of machines to factor the 232-digit number, RSA-768, and it took two years.

Prime factorization of common numbers

The following are the prime factorizations of some common numbers.

You can also email us on info calculat.io

Prime Factorization of 60

Prime Factors of

What is the Prime Factorization of 60?

Explanation of number 60 prime factorization.

Prime Factorization of 60 it is expressing 60 as the product of prime factors. In other words it is finding which prime numbers should be multiplied together to make 60.

Since number 60 is a Composite number (not Prime) we can do its Prime Factorization.

To get a list of all Prime Factors of 60, we have to iteratively divide 60 by the smallest prime number possible until the result equals 1.

Here is the complete solution of finding Prime Factors of 60:

The smallest Prime Number which can divide 60 without a remainder is 2 . So the first calculation step would look like:

60 ÷ 2 = 30

Now we repeat this action until the result equals 1 :

30 ÷ 2 = 15

Now we have all the Prime Factors for number 60. It is: 2, 2, 3, 5

Or you may also write it in exponential form: 2 2 × 3 × 5

Prime Factor Tree of 60

We may also express the prime factorization of 60 as a Factor Tree :

Prime factors of 60 factorization tree of 60

Related Calculations

  • Number 60 Facts
  • 60 in roman numerals
  • Factors of 60
  • 60 as a fraction
  • 60 in binary
  • Is 60 a prime number?

Prime factors of 60. Prime Factorization of 60. What is 60 as a product of Prime Factors?

Prime Factorization Calculator

What is a prime number, what is a prime factor, what is prime factorization, how to find prime factorization a factor tree method, greatest common factor, least common multiple, what is prime factorization of....

The prime factorization calculator will take any number and find its prime factors . Simply type the number into our tool and in no time you'll find the prime factorization. To understand the whole process, first you must get familiar with what is a prime factor. Once you understand that, we will move on to the difference between prime factor and prime factorization.

Below, you'll find all the answers, as well as concise information about how to find prime factorization and what a factor tree is.

To understand prime factorization, we need to start from the beginning - what is a prime number? A prime numbers are numbers whose only factors are one and itself - in other words, it can't be formed by multiplying two smaller natural numbers. A key point to note is that the two factors must be different, so 1 is not a prime number since both factors of 1 are the same. For example, 5 is a prime number since the only factors of 5 are 1 and 5. 6 is not a prime as, apart from 1 and 6, other factors exist - 2 and 3.

There are infinitely many primes and there's no simple formula to determine whether a number is a prime or not. That's why this prime factorization calculator is such a great, multi-purpose tool - you can use it as a prime number calculator as well!

Prime factors are factors of a number that are themselves prime numbers. For example, suppose we want to find the factors of 20, that is, we want to know what whole numbers multiply to give you 20. We know that 1 * 20 = 20 , 2 * 10 = 20 and 4 * 5 = 20 . But notice that 20, 10 and 4 are not prime factors. The only prime factors of 20 are 2 and 5 .

Prime factorization is when we break a number down into factors that are only prime numbers. If we look at the above example with 20, the factors are 1, 2, 4, 5, 10, 20 . The best place to start is to find at least one initial factor that is prime. Since 5 is prime, we can start with 4 * 5 . Notice that 4 is not prime, so we break 4 down into 2 * 2 . Since 2 is prime, the prime factorization of 20 is 2 * 2 * 5 . Go ahead and check this result with our prime factorization calculator.

We'll show you step by step how to find prime factorization. We'll use the factor tree diagram, an easy way to break down a number into its prime factors. Are you ready?

  • Take a number . There's no point in picking a prime number, as the prime factorization will finish at this point. Let's choose 36 .
  • Factor it into any two numbers , prime or not. You may want to take the easiest splits, e.g., if your number is even, split it into 2 and the other number. 36 is even so that we can write it as 2 * 18 .
  • Start constructing the factor tree . Draw two branches splitting down from your original number.

factor tree example, 36 split into 2 and 18

  • Factor the next line. If your number is a prime, leave it as it is. If it's not a prime number, repeat step 2.

factor tree, 36 split into 2 and 18, and 18 split into 2 and 9

  • Repeat step 4 until you're left with only prime numbers .

factor tree example, 36 split into 2 and 18, 18 into 2 and 9, 9 into 3 and 3

  • Write down the final prime factorization and prime factors .

full factor tree with leaves

Take all the "leaves" of your factor tree and multiply them together:

36 = 2 * 2 * 3 * 3

That's how we find prime factorization !

The prime factors of our original number 36 are 2 and 3 . The same prime factor may occur more than once, which is precisely what happened in our case - both prime numbers appear twice in the prime factorization. We can then express it as:

36 = 2² * 3²

Of course, you should get the same result with a different factor tree splits, as e.g. here:

factor tree, alternative split

Prime factorization is the first step in finding the greatest common factor - the greatest factor of two or more numbers. The GCF is especially useful for simplifying fractions and solving equations using polynomials. For example, the greatest common factor between 6 and 20 is 2 : prime factorization of the first number is 6 = 2 * 3 , the latter may be expressed as 20 = 5 * 2 * 2 , and the only number which appears in both prime factorizations is 2 , indeed. If you want to calculate greatest common factor in a snap, use our greatest common factor calculator .

The the prime factorization calculator is also useful in finding the least common multiple (LCM). The LCM is important when adding fractions with different denominators. The least common multiple is obtained when you multiply the higher powers of all factors between the two numbers. For example, the least common multiple between 6 and 20 is (2 * 2 * 3 * 5) = 60 . The LCM may be found by hand or with use of the least common multiple calculator .

Here is a list of prime factorizations for ease of reference. You can verify these with our prime factorization calculator.

8 = 2 × 2 × 2

11 is prime

12 = 2 × 2 × 3

13 is prime

16 = 2 × 2 × 2 × 2

17 is prime

18 = 2 × 3 × 3

19 is prime

20 = 2 × 2 × 5

22 = 2 × 11

23 is prime

24 = 2 × 2 × 2 × 3

26 = 2 × 13

27 = 3 × 3 × 3

28 = 2 × 2 × 7

29 is prime

30 = 2 × 3 × 5

31 is prime

32 = 2 × 2 × 2 × 2 × 2

33 = 3 × 11

34 = 2 × 17

36 = 2 × 2 × 3 × 3

37 is prime

38 = 2 × 19

39 = 3 × 13

40 = 2 × 2 × 2 × 5

41 is prime

42 = 2 × 3 × 7

43 is prime

44 = 2 × 2 × 11

45 = 3 × 3 × 5

46 = 2 × 23

47 is prime

48 = 2 × 2 × 2 × 2 × 3

50 = 2 × 5 × 5

51 = 3 × 17

52 = 2 × 2 × 13

53 is prime

54 = 2 × 3 × 3 × 3

55 = 5 × 11

56 = 2 × 2 × 2 × 7

57 = 3 × 19

58 = 2 × 29

59 is prime

60 = 2 × 2 × 3 × 5

61 is prime

62 = 2 × 31

63 = 3 × 3 × 7

64 = 2 × 2 × 2 × 2 × 2 × 2

65 = 5 × 13

66 = 2 × 3 × 11

67 is prime

68 = 2 × 2 × 17

69 = 3 × 23

70 = 2 × 5 × 7

71 is prime

72 = 2 × 2 × 2 × 3 × 3

73 is prime

74 = 2 × 37

75 = 3 × 5 × 5

76 = 2 × 2 × 19

77 = 7 × 11

78 = 2 × 3 × 13

79 is prime

80 = 2 × 2 × 2 × 2 × 5

81 = 3 × 3 × 3 × 3

82 = 2 × 41

83 is prime

84 = 2 × 2 × 3 × 7

85 = 5 × 17

86 = 2 × 43

87 = 3 × 29

88 = 2 × 2 × 2 × 11

89 is prime

90 = 2 × 3 × 3 × 5

91 = 7 × 13

92 = 2 × 2 × 23

93 = 3 × 31

94 = 2 × 47

95 = 5 × 19

96 = 2 × 2 × 2 × 2 × 2 × 3

97 is prime

98 = 2 × 7 × 7

99 = 3 × 3 × 11

100 = 2 × 2 × 5 × 5

101 is prime

102 = 2 × 3 × 17

103 is prime

104 = 2 × 2 × 2 × 13

105 = 3 × 5 × 7

108 = 2 × 2 × 3 × 3 × 3

117 = 3 × 3 × 13

120 = 2 × 2 × 2 × 3 × 5

121 = 11 × 11

125 = 5 × 5 × 5

126 = 2 × 3 × 3 × 7

130 = 2 × 5 × 13

132 = 2 × 2 × 3 × 11

135 = 3 × 3 × 3 × 5

140 = 2 × 2 × 5 × 7

144 = 2 × 2 × 2 × 2 × 3 × 3

147 = 3 × 7 × 7

150 = 2 × 3 × 5 × 5

162 = 2 × 3 × 3 × 3 × 3

175 = 5 × 5 × 7

180 = 2 × 2 × 3 × 3 × 5

196 = 2 × 2 × 7 × 7

200 = 2 × 2 × 2 × 5 × 5

210 = 2 × 3 × 5 × 7

216 = 2 × 2 × 2 × 3 × 3 × 3

225 = 3 × 3 × 5 × 5

245 = 5 × 7 × 7

250 = 2 × 5 × 5 × 5

256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

300 = 2 × 2 × 3 × 5 × 5

375 = 3 × 5 × 5 × 5

400 = 2 × 2 × 2 × 2 × 5 × 5

500 = 2 × 2 × 5 × 5 × 5

625 = 5 × 5 × 5 × 5

At the beginning, 1 was considered a prime number. It was not until the early 20th century that most mathematicians excluded 1 as a prime number. Notice the prime factorization calculator does not include 1 in the results of prime numbers.

Greatest common factor

Truncated cone volume.

  • Biology ( 97 )
  • Chemistry ( 98 )
  • Construction ( 143 )
  • Conversion ( 291 )
  • Ecology ( 29 )
  • Everyday life ( 259 )
  • Finance ( 564 )
  • Food ( 66 )
  • Health ( 438 )
  • Math ( 658 )
  • Physics ( 508 )
  • Sports ( 104 )
  • Statistics ( 182 )
  • Other ( 179 )
  • Discover Omni ( 40 )

Prime Factorization

Prime factorization is a way of expressing a number as a product of its prime factors . A prime number is a number that has exactly two factors, 1 and the number itself. For example, if we take the number 30. We know that 30 = 5 × 6, but 6 is not a prime number. The number 6 can further be factorized as 2 × 3, where 2 and 3 are prime numbers. Therefore, the prime factorization of 30 = 2 × 3 × 5, where all the factors are prime numbers.

Let us learn more about prime factorization with various mathematical problems followed by solved examples and practice questions.

What is Prime Factorization?

Prime factorization is the process of writing a number as the product of prime numbers. Prime numbers are the numbers that have only two factors, 1 and the number itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, and so on are prime numbers. Prime factorization of any number means to represent that number as a product of prime numbers. For example, the prime factorization of 40 can be done in the following way:

Prime Factorization of 40

Prime Factorization Meaning

The method of breaking down a number into its prime numbers that help in forming the number when multiplied is called prime factorization. In other words, when prime numbers are multiplied to obtain the original number, it is defined as the prime factorization of the number.

Prime Factorization of a Number

Let us see the prime factorization chart of a few numbers in the table given below:

What are Prime Factors?

The prime factors of a number are the 'prime numbers' that are multiplied to get the original number. For example, 2 and 5 are the prime factors of 20, i.e., 2 × 2 × 5 = 20. We know that the factors of a number are the numbers that are multiplied to get the original number. For example, 4 and 5 are the factors of 20, i.e., 4 × 5 = 20. Therefore, it should be noted that all the factors of a number may not necessarily be prime factors.

Prime factorization is similar to factoring a number but it considers only prime numbers (2, 3, 5, 7, 11, 13, 17, 19, and so on) as its factors. Therefore, it can be said that factors that divide the original number completely and cannot be split into more factors are known as the prime factors of the given number.

Methods of Prime Factorization

There are various methods for the prime factorization of a number. The most common methods that are used for prime factorization are given below:

  • Prime factorization by factor tree method
  • Prime factorization by division method

Prime Factorization by Factor Tree Method

In the factor tree method, the factors of a number are found and then those numbers are further factorized until we reach the prime numbers. Let us understand the prime factorization of a number using the factor tree method with the help of the following example.

Example: Do the prime factorization of 850 using the factor tree.

Solution: Let us get the prime factors of 850 using the factor tree given below.

  • Step 1: Place the number, 850, on top of the factor tree.
  • Step 2: Then, write down the corresponding pair of factors as the branches of the tree. Here, they are 25 and 34.
  • Step 3: Factorize the composite factors that are found in step 2, and write down the pair of factors as the next branches of the tree. Here, 25 can be further factorized into 5 × 5, and 34 can be factorized into 17 × 2
  • Step 4: Repeat step 3, until we get the prime factors of all the composite factors. So, we get the prime factors of 850 = 2 × 5 2 × 17

Prime Factorization by Factor Tree Method

Prime Factorization by Division Method

The division method can also be used to find the prime factors of a large number by dividing the number by prime numbers. Let us learn how to find the prime factors of a number by the division method using the following example.

Example: Do the prime factorization of 60 with the division method.

  • Step 1: Divide the number by the smallest prime number such that the smallest prime number should divide the number completely. Here we divide 60 by 2 to get 30.
  • Step 2: Again, divide the quotient of step 1 by the smallest prime number. So, 30 is again divided by 2 and we get 15.
  • Step 3: Repeat step 2, until the quotient becomes 1. Now, 15 is not divisible by 2, so we take the next prime number which is 3. And 15 ÷ 3 = 5. Then we divide 5 ÷ 5 = 1. Since we get 1 as the quotient, we stop here.
  • Step 4: Finally, multiply all the prime factors that are the divisors . Prime factorization of 60 = 2 × 2 × 3 × 5

Prime factorization by division method

Therefore, the prime factors of 60 are 2, 3, and 5.

Applications of Prime Factorization

Prime factorization is used extensively in the real world. The two most important applications of prime factorization are given below.

Cryptography and Prime Factorization

Hcf and lcm using prime factorization.

Cryptography is a method of protecting information using codes. Prime factorization plays an important role for the coders who create a unique code using numbers which is not too heavy for computers to store or process quickly.

To find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers, we use the prime factorization method. For this, we first do the prime factorization of both the numbers. The following points related to HCF and LCM need to be kept in mind:

  • HCF is the product of the common prime factors with the smallest powers.
  • LCM is the product of the common prime factors with the highest powers

Example: What is the HCF and LCM of 850 and 680?

Solution: We will first do the prime factorization of both the numbers.

  • The prime factorization of 850 is: 850 = 2 1 × 5 2 × 17 1
  • The prime factorization of 680 is: 680 = 2 3 × 5 1 × 17 1
  • Observing this, we can see that the common prime factors of 850 and 680 with the smallest powers are 2 1 , 5 1 and 17 1 , and the common prime factors with the highest powers are 2 3 , 5 2 , 17 1
  • HCF is the product of the common prime factors with the smallest powers. Hence, HCF of (850, 680) = 2 1 × 5 1 × 17 1 = 170
  • LCM is the product of the common prime factors with the highest powers. Hence, LCM of (850, 680) = 2 3 × 5 2 × 17 1 = 3400
  • Thus, HCF of (850, 680) = 170, LCM of (850, 680) = 3400

☛ Related Articles

  • Prime Factorization of 60
  • Prime Factorization of 36
  • Prime Factorization of 30
  • Prime Factorization of 64
  • Prime Factorization of 45
  • Prime Factorization of 50
  • Prime Factorization of 48
  • Prime Factorization of 40
  • Prime Factorization of 8
  • Prime Factorization of 24
  • Prime Factorization of 12
  • Prime Factorization of 72
  • Prime Factorization of 18
  • Prime Factorization of 42

Prime Factorization Examples

Example 1: Express 1080 as the product of prime factors.

We will do the prime factorization of 1080 as follows:

Prime Factorization of 1080

Thus, 1080 = 2 3 × 3 3 × 5

Therefore, the prime factorization of 1080 is 2 3 × 3 3 × 5

Example 2: Find the lowest common multiple of 48 and 72 using prime factorization.

We will do the prime factorization of 48 and 72 as shown below:

Prime Factorization of 48

The prime factorization of 72 is shown below:

Prime Factorization of 72

So, the prime factors of 48 = 2 4 × 3 1

the prime factors of 72 = 2 3 × 3 2

Observing this we can see that the common prime factors of 48 and 72 with the greatest powers are 2 4 , 3 2

The LCM of any 2 numbers is the product of the common prime factors with the greatest powers. Hence, LCM (48, 72) = 2 4 × 3 2 = 144

Therefore, LCM (48, 72) = 2 4 × 3 2 = 144

Example 3: Show the prime factorization of 40 using the division method and the factor tree method.

Let us use the division method and the factor tree method to prove that the prime factorization of 40 will always remain the same.

Prime Factorization of 40

Therefore, this shows that by any method of factorization, the prime factorization remains the same. The prime factorization for a number is unique.

go to slide go to slide go to slide

how to write 60 as a product of prime factors

Book a Free Trial Class

Practice Questions on Prime Factorization

go to slide go to slide

FAQs on Prime Factorization

What is prime factorization in math.

Prime factorization of any number means to represent that number as a product of prime numbers. A prime number is a number that has exactly two factors, 1 and the number itself. For example, the prime factorization of 18 = 2 × 3 × 3. Here 2 and 3 are the prime factors of 18.

How to do Prime Factorization?

Prime factorization of any number can be done by using two methods:

  • Division method - In this method, the given number is divided by the smallest prime number which divides it completely. After this, the quotient is again divided by the smallest prime number. This step is repeated until the quotient becomes 1. Then, all the prime factors that are divisors are multiplied.
  • Factor tree method - In this method, the given number is placed on top of the factor tree. Then, the corresponding pairs of factors are written as the branches of the tree. After this step, the composite factors are again factorized and written down as the next branches. This procedure is repeated until we get the prime factors of all the composite factors. A detailed explanation of both these methods, with examples, is given above on this page.

What are Prime Factors of a Number?

The prime factors of a number are the 'prime numbers' that are multiplied to get the original number. For example, 2 and 3 are the prime factors of 12, i.e., 2 × 2 × 3 = 12. It can also be said that factors that divide the original number completely and cannot be split further into more factors are known as the prime factors of the given number. It should be noted that 4 and 6 are also factors of 12 but they are not prime numbers, therefore, we do not write them as prime factors of 12.

What is the Prime Factorization of 72, 36, and 45?

Prime factorization is the way of writing a number as the multiple of their prime factors. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and so on. The prime factorization of 72, 36, and 45 are shown below.

  • Prime factorization of 72 = 2 3 × 3 2
  • Prime factorization of 36 = 2 2 × 3 2
  • Prime factorization of 45 = 3 2 × 5

How to Find LCM using Prime Factorization?

The abbreviation LCM stands for 'Least Common Multiple'. The Least Common Multiple (LCM ) of a number is the smallest number that is the product of two or more numbers. The LCM of two numbers can be calculated by first finding out the prime factors of the numbers. The LCM is the product of the common prime factors with the greatest powers. For example, let us find the LCM of 12 and 18. The prime factorization of 12 = 2 2 × 3 1 , and the prime factorization of 18 = 2 1 × 3 2 . Among the common prime factors, the product of the factors with the highest powers is 2 2 × 3 2 = 36.

How to Find HCF using Prime Factorization?

The abbreviation HCF stands for 'Highest Common Factor'. The Highest Common Factor (HCF) of two numbers is the highest possible number which divides both the numbers completely. The HCF of two numbers can be found out by first finding out the prime factors of the numbers. The HCF is the product of the common prime factors with the smallest powers. For example, let us find the HCF of 12 and 18. The prime factorization of 12 = 2 2 × 3 1 , and the prime factorization of 18 = 2 1 × 3 2 . Among the common prime factors, the product of the factors with the smallest powers is 2 1 × 3 1 = 6.

Why is Prime Factorization Important?

Prime factorization is used to find the HCF and LCM of numbers. It is widely used in cryptography which is the method of protecting information using codes. Prime numbers are used to form or decode those codes.

What is the Prime Factorization of 24?

The number 24 can be written as 4 × 6. Now the composite numbers 4 and 6 can be further factorized as 4 = 2 × 2 and 6 = 2 × 3. Therefore, the prime factorization of 24 is 24 = 2 × 2 × 2 × 3 = 2 3 × 3

How is Prime Factorization used in the Real World?

Prime factorization is used extensively in the real world. For example, if we need to divide anything into equal parts, or we need to exchange money, or calculate the time while travelling, we use prime factorization. One common example is, if we have 21 candies and we need to divide it among 3 kids, we know the factors of 21 as, 21 = 3 × 7. This means we can distribute 7 candies to each kid.

When to use Prime Factorization?

Prime factorization is one of the methods used to find the Greatest Common Factor (GCF) of a given set of numbers. GCF by prime factorization is useful for larger numbers for which listing all the factors is time-consuming.

How to Find Prime Factors of a Number?

The prime factors of a number can be listed using various methods. It should be noted that prime factors are different from factors because prime factors are prime numbers that are multiplied to get the original number. One of the methods to find the prime factors of a number is the division method. Let us use this method to find the prime factors of 24.

  • In this method, the given number is divided by the smallest prime number which divides it completely. So, 24 ÷ 2 = 12.
  • After this, the quotient is again divided by the smallest prime number. So 12 ÷ 2 = 6.
  • This step is repeated until the quotient becomes 1. This means 6 ÷ 2 = 3. Now 3 cannot be further divided or factorized because it is a prime number.
  • Then, all the prime factors that are divisors are multiplied and listed.
  • So we get 24 = 2 × 2 × 2 × 3 and we know that the prime factors of 24 are 2 and 3 and the prime factorization of 24 = 2 3 × 3

What is the Definition of Prime Factorization?

Prime factorization is defined as the way of expressing a number as a product of its prime factors. We know that a prime number is a number that has exactly two factors, 1 and the number itself. For example, if we take the number 20. We know that 20 = 5 × 4, but 4 is not a prime number. The number 4 can further be factorized as 2 × 2, where 2 is a prime number. Therefore, the prime factorization of 20 = 2 × 2 × 5, where all the factors are prime numbers.

  • 90% Refund @Courses
  • School Guide
  • Mathematics
  • Number System and Arithmetic
  • Trigonometry
  • Probability
  • Mensuration
  • Maths Formulas
  • Class 8 Maths Notes
  • Class 9 Maths Notes
  • Class 10 Maths Notes
  • Class 11 Maths Notes
  • Class 12 Maths Notes

Related Articles

  • Explore Our Geeks Community
  • What is Arithmetic?
  • How many ways can you roll a sum of 8 with two dice?
  • Difference between Natural and Whole numbers
  • Simplify 2m 2 ×(2m 3 )
  • Why diagonals of rhombus are not equal?
  • Mention any 4 rational numbers which are less than 5
  • Is there a smallest number and a largest number?
  • In a triangle ABC right angled at B, find the value of tan A
  • Ratio Formula
  • What is larger than infinity?
  • Find a simple expression for e {In x - 2ln y}
  • Find the square of: 3a+7b
  • Find the exact value of csc 270°
  • If f(x) = 3x 2 - 2x + 4 and g(x) = 2 - 3x, then find f(x)g(x) for x = -1
  • Relative Minima and Maxima
  • Find the square of: 2a+b
  • What is the common difference of the AP 1/b, (3-b)/3b, (3-2b)/3b...?
  • Find the probability that the difference of the points on the dice is 2 or 3 when two dice are thrown simultaneously
  • What is the probability of rolling the same number on 3 dice?

Factors of 60

Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 . Factors of 60, is a set of numbers that perfectly divides 60. Factors of 60 are the numbers that divide 60 without leaving any remainder (i.e., with a remainder = 0).

What are Factors of 60

In this article, we will learn about, Factors, Factors of 60 along with Prime Factorization and Factor Tree of 60.

Table of Content

How To Find Factors of 60?

Prime factorization of 60, factor tree of 60.

  • Factor Pairs

What are Factors?

Factors are number that divides the given number evenly or perfectly i.e. Factors of a number completely divides that number without leaving any reminder behind.

Factors of a number can also be defined as the numbers which when multiplied in pairs returns the original number. Examples of Factors are:

  • Factors of 15: 1, 3, 5 and 15.
  • Factors of 18: 1, 2, 3, 6, 9, and 18.

Note : Every number has 1 and number itself as its factor.

What are factors of 60.

Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20 ,30 and 60. Factors of 60 according to its definition are the numbers that can be multiplied with each other to give the product 60 as a result. Factors of 60 can be represented as:

1 × 60 = 60 2 × 30 = 60 3 × 20 = 60 4 × 15 = 60 5 × 12 = 60 6 × 10 = 60 Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Also, Each of this factor when divided by 60 leaves back the remainder as 0.

Read More,  Factors of a number .

All Factors of 60

Here is a list of all the factors of 60:

Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Prime Factors of 60

Prime factors are prime numbers which when multiplied together gives back the original number. In simple words, prime factors are set of prime numbers which can divide it until the original number become 1.

Prime Factor of 60 are 2 , 3 and 5.

But we express it in given form:

Prime Factor of 60 are: 2 2 ,3 and 5 which are expressed as 2 2 × 3 × 5 = 60.
prime numbers  Whole number

In order to find the factors of 60, we need to identify all numbers that can divide 60 without leaving any remainder.

Here are steps we can follow:

  • Strat from 1 till 60 and check if the number can divide 60 without leaving any reminder.
  • If yes, then note down both the number and the result when 60 is divided by it.
  • Else, check for next number and repeat it till number reaches 60.
  • List all the Factors and the resulting list is 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

To check that all number obtained are satisfying the definition of factors that a factor can divide number without leaving any reminder:

The above table provides, the factors of 60 as 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Prime Factorization is a method of obtaining factors of any number by dividing it with a prime number until the quotient changes to 1. Prime factorization is used to express a number as the product of its primes.

To find the prime factorization of 60, follow the given steps:

  • Step 1: Choose smallest prime number(which is 2 here) which can perfectly divide 60 .
  • Step 2: Divide 60 with the chosen number and note the result(30) and factor which is 2.
  • Step 3: Repeat again same steps until the quotient turns to 1.
  • Step 4: List all the numbers together to get all the prime factors of 60.

Below is the representation of Prime Factorization of 60.

Prime-Factors-of-60

Read more about Prime Factorization .

Factor Tree refers to the representation of a number as a product of its primes in the form of branches and leaves. A factor tree is a diagram that divides a number into its prime factors and represent it in a tree form.

The steps to draw prime factorization factor tree of 60 are as follows:

Step 1: Start with 60. Step 2: Now find the smallest prime factor that divides 60. It’s 2 as shown in figure for first branch under 60. Step 3: Continue dividing each branch by prime factors until you reach only prime numbers Step 4: The prime factors are now at the ends of each branch. Arrange the list in ascending order, the prime factorization of 60 = 2 × 2 × 3 × 5. Step 5: Exponents can be used to express repeated prime factors in form 2² × 3 × 5 for 60.

Here’s a factor tree for the number 60:

Factor-Tree-of-60

Factor Pairs of 60

As you might have noticed in above list of equation that result after dividing a number gives another factor. A factor pair of a number is the set of two of its factors, such that when multiplied gives number itself. In simple mathematics words, when we multiply two numbers we get a product. The factors of this product are the numbers that were multiplied to obtain it. Factor pairs refer to two numbers that, when multiplied together to gain a particular product.

As negative number when multiplied with another negative number it gives positive number so, here Negative numbers pairs can also be considered, So we can divide factor pairs on the basis of Positive factor pair and Negative factor pair.

Positive Factor Pairs of 60

Positive factor pairs of 60 are the pair of positive integer whose product gives 60 as a result.

Negative Factor Pairs of 60

Negative factor pairs of 60 are the pair of negative integer whose product gives 60(positive 60) as a result.

Factors of 12 Factors of 36

Factors of 60 – Solved Examples

Example 1: What is the product of all the factors of 60?

The factors of 60 are 1, 2, 3, 4, 6, 10, 12, 15 ,20, 30 and 60. So, product =1 × 2 × 3 × 4 × 5 × 6 × 10 × 12 × 15 × 20 × 30 × 60. = 46656000000.

Example 2: What is the largest possible factor of 60 other than 60?

30 is the largest possible factor of 60 other than 60.

Example 3: What are the prime factors of 60?

Prime factor of 60 are 2 2 , 3 and 5.

Example 4: If d is a factor of both 60 and 15, what are the possible values of d?

Factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15 ,20, 30 and 60. Factors of 15 are: 1, 3, 5 and 15. So possible values of d are: 1, 3, 5 and 15.

Example 5: What are the common factors of 60 and 45?

Factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15 ,20, 30 and 60. Factors of 45 are: 1, 3, 5, 9, 15, and 45. Common Factors of 60 and 45 are : 1, 3, 5 and 15.

Factors of 60 – Practice Questions

Q1: Is (-2, -30) a negative factor pair of 60?

Q2: Write all factors of 60.

Q3: What is sum of all factors of 60?

Q4: Is 15 a factor of 60?

Q5: Is 60 itself a factor of 60?

Factor of 60: Frequently Asked Questions

What are the factors of 60.

Factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15 ,20, 30 and 60.

Can Negative Numbers be Considered as Factors of 60?

Yes, negative numbers can be considered as Factors of 60.

What are the Possible Negative Factors of 60?

-1,- 2, -3,- 4,- 5,- 6, -10,-12, -15 ,-20, -30 and -60.

60 is Factor or Multiple of 60?

60 is both a factor and a multiple of 60.

Is 1 and Number itself a Factor of Every Number?

Yes , 1 and the number itself is always a factor of itself.

Whether you're preparing for your first job interview or aiming to upskill in this ever-evolving tech landscape, GeeksforGeeks Courses are your key to success. We provide top-quality content at affordable prices, all geared towards accelerating your growth in a time-bound manner. Join the millions we've already empowered, and we're here to do the same for you. Don't miss out - check it out now !

Looking for a place to share your ideas, learn, and connect? Our Community portal is just the spot! Come join us and see what all the buzz is about!

Please Login to comment...

  • Geeks Premier League 2023
  • Math-Concepts
  • Geeks Premier League
  • School Learning
  • School Mathematics
  • 10 Best AI Tools for Banner Designing in 2024
  • Google Bard Advanced AI: A Multimodal Chatbot Competitor Emerges
  • 12 Best AI Tools for Interior Designs in 2024
  • How to Automate Data Entry Tasks with ChatGPT
  • 30 OOPs Interview Questions and Answers (2024)

Improve your Coding Skills with Practice

 alt=

What kind of Experience do you want to share?

Spark Academy

  • Student Login
  • Student Workshops

></center></p><h2>60 as a Product of Prime Factors</h2><p>I’m here to unravel the mystery of 60 and its prime factors. Let’s dive into the world of numbers and explore how 60 can be expressed as a product of prime factors.</p><p>So, what are prime factors? They are the building blocks of a number, the prime numbers that multiply together to create it. In the case of 60, its prime factors are 2, 2, 3, and 5.</p><p>Imagine these prime factors as puzzle pieces. When we put them together, we get the complete picture of 60. It’s like assembling a jigsaw puzzle, where each prime factor represents a unique piece. The final answer is 60 = 5 x 3 x 2 x 2.</p><p>But why stop there? Let’s explore more about 60. It’s not just a whole number, but also a composite number, meaning it can be divided evenly by multiple factors. In fact, the factors of 60 include 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. These factors can be paired up, like (1, 60), (2, 30), (3, 20), and so on. And here’s an interesting tidbit: the product of these paired factors is always 60.</p><p>To sum it all up, 60 is a fascinating number with a prime factorization of 2 × 2 × 3 × 5. Its factors can be found using a factor tree or by pairing them up, and their product is always 60. So, whether you’re a math enthusiast or simply curious about numbers, 60 and its prime factors offer a captivating journey into the world of mathematics.</p><h2>Key Takeaways</h2><p>The prime factorization of 60 is 2 × 2 × 3 × 5 or 22 × 3 × 5.</p><p>The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.</p><p>The factors of 60 can be written in pairs, such as (1, 60), (2, 30), (3, 20), etc.</p><p>The product of the pair factors of 60 is always 60.</p><p>The sum of all the factors of 60 is 168.</p><h2>Understanding 60 as a Product of Prime Factors</h2><p>When it comes to understanding numbers, breaking them down into their prime factors can provide valuable insight. In this section, we will explore the prime factorization of the number 60 and how it can be expressed as a product of its prime factors.</p><h2>What are Prime Factors?</h2><p>Before we delve into the prime factorization of 60, let’s first understand what prime factors are. Prime factors are the prime numbers that, when multiplied together, give us a composite number. A composite number is a whole number that can be divided evenly by numbers other than 1 and itself. In the case of 60, we will discover the prime factors that, when multiplied, result in this composite number.</p><h2>Prime Factorization of 60</h2><p>The prime factorization of 60 reveals the prime factors that make up this number. By determining the prime factors, we can express 60 as a product of these primes. In the case of 60, the prime factors are 2, 2, 3, and 5.</p><p>To express 60 as a product of its prime factors, we can write it as 2 × 2 × 3 × 5 or 22 × 3 × 5. The order in which the prime factors are written does not matter, as multiplication is commutative. Therefore, both expressions are correct.</p><p>By breaking down 60 into its prime factors, we gain a deeper understanding of its composition and can use this knowledge in various mathematical operations. The prime factorization not only allows us to express 60 in its simplest form but also helps us find other properties of the number.</p><h2>Exploring the Factors of 60</h2><p>In addition to its prime factorization, it’s useful to examine the factors of 60. Factors are the numbers that divide evenly into a given number. For 60, the factors include 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.</p><p>The factors of 60 can be written in pairs, such as (1, 60), (2, 30), (3, 20), and so on. These pairs represent the numbers that, when multiplied together, yield 60. It’s interesting to note that regardless of the pair chosen, their product will always equal 60.</p><h2>Using a Factor Tree and Summing the Factors</h2><p>There are various methods to find the factors of a number, and one such method is using a factor tree. By repeatedly dividing a number into its prime factors, we can construct a factor tree that displays all the factors.</p><p>Another interesting property of 60 is the sum of its factors. By adding up all the factors, we find that the sum of the factors of 60 is 168. This sum provides insight into the overall magnitude of the factors and can be useful in certain mathematical calculations.</p><p>In conclusion, understanding the prime factorization of a number like 60 allows us to express it as a product of its prime factors and gain a deeper understanding of its composition. The factors of 60 can be explored through pairs, a factor tree, or by summing them. By utilizing these techniques, we can unlock the hidden properties of numbers and apply them to various mathematical concepts.</p><p>Reference:  Express 60 as a Product of its Prime Factors</p><h2>The Factors of 60</h2><p>As a skilled content creator, I have come across a fascinating mathematical concept that I would love to share with you today – the factors of 60. Understanding the factors of a number is not only important in mathematics but also has practical applications in various fields, such as finance, engineering, and computer science. So, let’s dive into the world of factors and explore the intriguing properties of the number 60!</p><h2>Finding Factors</h2><p>To begin our exploration, we first need to understand how to find the factors of a number. Factors are the numbers that can be evenly divided into a given number without leaving a remainder. In the case of 60, we are interested in finding all the numbers that divide 60 without a remainder.</p><h2>Pairs of Factors</h2><p>One interesting aspect of the factors of 60 is that they can be written in pairs. These pairs consist of two numbers whose product equals 60. For example, the factors of 60 include the pairs (1, 60), (2, 30), (3, 20), and so on. It’s like finding puzzle pieces that fit together perfectly to form the number 60.</p><h2>Sum of Factors</h2><p>Now, let’s consider the sum of all the factors of 60. When we add up all the factors, we get a surprising result – the sum is 168! This means that if we were to gather all the factors of 60 together, their combined value would be 168. It’s fascinating how these seemingly unrelated numbers come together to create such a specific sum.</p><p>Behind the scenes, the factors of 60 are actually derived from its prime factors. Prime factors are the prime numbers that, when multiplied together, give us the original number. In the case of 60, its prime factors are 2, 2, 3, and 5. By multiplying these prime factors, we obtain the product 60. So, we can express 60 as a product of its prime factors: 2 × 2 × 3 × 5 or 22 × 3 × 5.</p><p>If you’d like to explore this topic further, you can check out this  source  that I came across during my research. It provides additional information and examples on expressing 60 as a product of its prime factors.</p><p>In conclusion, understanding the factors of a number, such as 60, not only enhances our mathematical knowledge but also has practical applications in various fields. By finding the factors, exploring the pairs they form, and discovering their sum, we gain valuable insights into the properties of numbers. So, let’s continue our journey of mathematical exploration and uncover the wonders hidden within numbers.</p><h2>Using a Factor Tree for 60</h2><p>Have you ever wondered how to break down a number into its prime factors? Well, today I’m here to show you how to do just that for the number 60! By using a factor tree, we can easily find the prime factors of 60 and express it as a product of those factors.</p><h2>Step-by-Step Process</h2><p>Let’s dive right into the step-by-step process of finding the prime factors of 60 using a factor tree.</p><ul><li>Start by writing the number 60 at the top of your paper.</li><li>Think of two factors of 60 that multiply together to give you 60. In this case, we can choose 2 and 30.</li><li>Now, let’s break down those factors further. 30 can be factored into 2 and 15.</li><li>Keep breaking down the factors until you can no longer divide them. For 15, we have 3 and 5.</li><li>Now, let’s look at the other branch of the factor tree. 2 cannot be further divided, so we keep it as is.</li><li>Write down all the prime factors you have found: 2, 2, 3, and 5.</li><li>Finally, express 60 as a product of these prime factors: 60 = 2 x 2 x 3 x 5.</li></ul><p>By following this step-by-step process, we have successfully found the prime factors of 60 and expressed it as a product. But what exactly are prime factors?</p><h2>Finding Prime Factors</h2><p>Prime factors are the prime numbers that multiply together to give you a certain number. In the case of 60, the prime factors are 2, 2, 3, and 5. These are the smallest prime numbers that, when multiplied, equal 60.</p><p>The prime factorization of 60 can be written as 2 × 2 × 3 × 5 or 22 × 3 × 5. It’s important to note that the order in which we write the prime factors doesn’t matter. The result will be the same.</p><p>Now that we know the prime factors of 60, let’s take a look at some other interesting properties of this number.</p><p>According to  Cuemath , 60 is not only a whole number but also a composite number. This means that 60 has factors other than 1 and itself. In fact, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.</p><p>One fascinating aspect of the factors of 60 is that they can be written in pairs. For example, we have the pairs (1, 60), (2, 30), (3, 20), and so on. If you multiply the numbers in each pair, you’ll always get 60 as the product. Isn’t that interesting?</p><p>Another way to find the factors of 60 is by using a factor tree, as we discussed earlier. The factor tree method provides a visual representation of the factors and their relationships, making it easier to understand the concept.</p><p>In conclusion, the prime factors of 60 are 2, 2, 3, and 5. By breaking down 60 into its prime factors, we can express it as a product, which is 60 = 2 x 2 x 3 x 5. The factors of 60 can be found using a factor tree or by writing them in pairs. So go ahead and give it a try yourself!</p><h2>Properties of 60 as a Product of Prime Factors</h2><p>In mathematics, numbers can often be expressed as a product of prime factors. Let’s take a closer look at the prime factorization of the number 60.</p><h2>Composite Number</h2><p>Firstly, it is important to note that 60 is a composite number. A composite number is a positive integer that has factors other than 1 and itself. In other words, it can be divided evenly by numbers other than 1 and 60.</p><h2>Whole Number</h2><p>Additionally, 60 is considered a whole number. Whole numbers are non-negative integers that include zero and all positive integers without any fractional or decimal parts.</p><p>Now, let’s dive into the prime factorization of 60.</p><p>The prime factors of 60 are 2, 2, 3, and 5. We can express the prime factorization of 60 as 2 × 2 × 3 × 5 or 22 × 3 × 5. This means that when we multiply these prime factors together, we get the original number 60.</p><h2>Factors of 60</h2><p>The factors of 60 are the numbers that can be evenly divided into 60 without leaving a remainder. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. These factors can be paired up, such as (1, 60), (2, 30), (3, 20), and so on.</p><h2>Factor Tree</h2><p>Another way to find the factors of 60 is by using a factor tree. A factor tree is a diagram that breaks down a number into its prime factors. By using a factor tree, we can easily visualize and identify the prime factors of 60.</p><p>The sum of all the factors of 60 is 168. This means that if we add up all the factors of 60, the resulting sum is 168.</p><p>For a more detailed explanation and examples, you can refer to  this source .</p><p>By understanding the properties of 60 as a product of prime factors, we can gain insight into its divisibility and relationships with other numbers.</p><h2>Applications of Prime Factorization</h2><p>Prime factorization is a fundamental concept in number theory that involves expressing a number as a product of its prime factors. This process has various applications in mathematics and other fields. In this section, we will explore two practical applications of prime factorization: simplifying fractions and finding the greatest common factor (GCF) and least common multiple (LCM) of numbers.</p><h2>Simplifying Fractions</h2><p>One useful application of prime factorization is simplifying fractions. When we have a fraction with a numerator and denominator that share common factors, we can simplify the fraction by canceling out these factors. Prime factorization helps us identify these common factors efficiently.</p><p>Let’s consider an example: the fraction 24/36. To simplify this fraction, we can find the prime factors of both the numerator and denominator. The prime factorization of 24 is 2 x 2 x 2 x 3, while the prime factorization of 36 is 2 x 2 x 3 x 3. By canceling out the common factors, we obtain 1/3 as the simplified form of the fraction.</p><h2>Finding GCF and LCM</h2><p>Another practical application of prime factorization is finding the greatest common factor (GCF) and least common multiple (LCM) of numbers. The GCF is the largest number that divides two or more numbers without leaving a remainder, while the LCM is the smallest multiple that is divisible by two or more numbers.</p><p>Let’s illustrate this application using the number 60. We can express 60 as a product of its prime factors: 2 x 2 x 3 x 5. The prime factors of 60 are 2, 2, 3, and 5. By comparing the prime factors of different numbers, we can determine their GCF and LCM.</p><p>For example, let’s find the GCF of 60 and 90. The prime factorization of 90 is 2 x 3 x 3 x 5. By comparing the common prime factors of 60 and 90, we find that the GCF is 2 x 3 x 5, which is 30.</p><p>Similarly, we can find the LCM of 60 and 90 by considering their prime factorizations. The LCM is obtained by taking the highest power of each prime factor that appears in either number. In this case, the LCM of 60 and 90 is 2 x 2 x 3 x 3 x 5, which equals 180.</p><p>Prime factorization provides a systematic approach to finding the GCF and LCM of numbers, making it a valuable tool in various mathematical calculations.</p><p>In conclusion, prime factorization has practical applications in simplifying fractions and finding the GCF and LCM of numbers. By expressing numbers as products of their prime factors, we can simplify fractions, determine common factors, and find the smallest or largest divisible numbers. Understanding prime factorization enhances our problem-solving abilities and helps us tackle complex mathematical problems efficiently.</p><p>For more information and examples on prime factorization, you can refer to this  article .</p><h2>Examples of Prime Factorization</h2><p>Prime factorization is the process of breaking down a number into its prime factors. It provides a unique representation of a number as a product of its prime factors. In this section, we will explore an example of prime factorization for the number 60.</p><p>To find the prime factorization of 60, we need to determine the prime numbers that can divide 60 evenly. The prime factors of 60 are 2, 2, 3, and 5. We can express 60 as the product of these prime factors:</p><p>60 = 2 x 2 x 3 x 5</p><p>Alternatively, we can write it as:</p><p>60 = 2^2 x 3 x 5</p><p>This representation shows that 60 is composed of two 2’s, one 3, and one 5.</p><h2>Prime Factorization of Other Numbers</h2><p>Prime factorization can be applied to any whole number. By breaking down a number into its prime factors, we can gain insights into its properties and relationships with other numbers.</p><p>For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. These factors can be written in pairs, such as (1, 60), (2, 30), (3, 20), and so on. It’s interesting to note that the product of the pair factors of 60 is always 60.</p><p>Another method to find the factors of 60 is by using a factor tree. Starting with the number 60, we can branch out and divide it into its prime factors until we reach the prime numbers. The branches of the factor tree represent the prime factors, and the product of all the branches gives us the original number.</p><p>Furthermore, the sum of all the factors of 60 is 168. This sum includes both the pair factors and the individual factors of 60.</p><p>In conclusion, prime factorization is a useful tool for understanding the composition of a number and its relationships with other numbers. By expressing a number as a product of its prime factors, we can gain insights into its properties and characteristics. The prime factorization of 60, for example, reveals that it is composed of the prime numbers 2, 2, 3, and 5. Through examples like these, we can explore the fascinating world of prime factorization and its applications in mathematics.</p><p>In conclusion, 60 can be expressed as a product of its prime factors. The prime factors of 60 are 2, 2, 3, and 5. By multiplying these prime factors together, we get the final answer of 60 = 5 x 3 x 2 x 2.</p><p>As we have learned, 60 is a whole number and a composite number. Its prime factorization can be represented as 2 × 2 × 3 × 5 or 22 × 3 × 5. This means that 60 is divisible by these prime numbers only.</p><p>The factors of 60 are the numbers that divide evenly into 60. These include 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. These factors can be written in pairs, such as (1, 60), (2, 30), (3, 20), and so on. It is interesting to note that the product of any pair of factors of 60 is always equal to 60.</p><p>Another way to find the factors of 60 is by using a factor tree. This method involves breaking down the number into its prime factors and combining them to find all possible factors.</p><p>In addition, the sum of all the factors of 60 is 168. This means that when we add up all the factors of 60, we get a total of 168.</p><p>To learn more about expressing 60 as a product of its prime factors, you can refer to  this helpful resource .</p><p>In this section, we have explored the prime factorization, factors, and sum of factors of 60. Understanding these concepts can be useful in various mathematical calculations and problem-solving scenarios.</p><h2>Frequently Asked Questions</h2><p>What are the prime factors of 60.</p><p>The prime factors of 60 are 2, 2, 3, and 5.</p><h2>How can 60 be expressed as a product of prime factors?</h2><p>Is 60 a whole number.</p><p>Yes, 60 is a whole number.</p><h2>Is 60 a composite number?</h2><p>Yes, 60 is a composite number as it has factors other than 1 and itself.</p><h2>What are the factors of 60?</h2><p>How can the factors of 60 be written in pairs, what is the product of the pair factors of 60, can the factors of 60 be found using a factor tree.</p><p>Yes, the factors of 60 can also be found using a factor tree.</p><h2>What is the sum of all the factors of 60?</h2><p>Looking for maths help  get in touch here..</p><ul><li>Home   |  </li><li>About   |  </li><li>Contact Us   |  </li><li>Privacy   |  </li><li>Copyright   |  </li><li>Shop   |  </li><li>🔍 Search Site</li><li>Kindergarten</li><li>All Generated Sheets</li><li>Place Value Generated Sheets</li><li>Addition Generated Sheets</li><li>Subtraction Generated Sheets</li><li>Multiplication Generated Sheets</li><li>Division Generated Sheets</li><li>Money Generated Sheets</li><li>Negative Numbers Generated Sheets</li><li>Fraction Generated Sheets</li><li>Place Value Zones</li><li>Number Bonds</li><li>Addition & Subtraction</li><li>Times Tables</li><li>Fraction & Percent Zones</li><li>All Calculators</li><li>Fraction Calculators</li><li>Percent calculators</li><li>Area & Volume Calculators</li><li>Age Calculator</li><li>Height Calculator</li><li>Roman Numeral Calculator</li><li>Coloring Pages</li><li>Fun Math Sheets</li><li>Math Puzzles</li><li>Mental Math Sheets</li><li>Online Times Tables</li><li>Online Addition & Subtraction</li><li>Math Grab Packs</li><li>All Math Quizzes</li><li>1st Grade Quizzes</li><li>2nd Grade Quizzes</li><li>3rd Grade Quizzes</li><li>4th Grade Quizzes</li><li>5th Grade Quizzes</li><li>6th Grade Math Quizzes</li><li>Place Value</li><li>Rounding Numbers</li><li>Comparing Numbers</li><li>Number Lines</li><li>Prime Numbers</li><li>Negative Numbers</li><li>Roman Numerals</li><li>Subtraction</li><li>Add & Subtract</li><li>Multiplication</li><li>Fraction Worksheets</li><li>Learning Fractions</li><li>Fraction Printables</li><li>Percent Worksheets & Help</li><li>All Geometry</li><li>2d Shapes Worksheets</li><li>3d Shapes Worksheets</li><li>Shape Properties</li><li>Geometry Cheat Sheets</li><li>Printable Shapes</li><li>Coordinates</li><li>Measurement</li><li>Math Conversion</li><li>Statistics Worksheets</li><li>Bar Graph Worksheets</li><li>Venn Diagrams</li><li>All Word Problems</li><li>Finding all possibilities</li><li>Logic Problems</li><li>Ratio Word Problems</li><li>All UK Maths Sheets</li><li>Year 1 Maths Worksheets</li><li>Year 2 Maths Worksheets</li><li>Year 3 Maths Worksheets</li><li>Year 4 Maths Worksheets</li><li>Year 5 Maths Worksheets</li><li>Year 6 Maths Worksheets</li><li>All AU Maths Sheets</li><li>Kindergarten Maths Australia</li><li>Year 1 Maths Australia</li><li>Year 2 Maths Australia</li><li>Year 3 Maths Australia</li><li>Year 4 Maths Australia</li><li>Year 5 Maths Australia</li><li>Meet the Sallies</li><li>Certificates</li></ul><h2>Prime Factorization Calculator</h2><p><center><img style=

Welcome to our Prime Factorization Calculator page. This calculator will convert a number into its unique product of prime factors.

You can also use the calculator for finding all the factors of any given number.

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

Prime Factorization Calculator - How to use it

Input any number values into our Prime Factorization Calculator and it will find quickly find all of the factors and then re-write the number as a unique product of its prime factors.

It will also show you what the number looks like in exponential form.

Note: prime numbers cannot be written as a product of prime factors as their only factors are 1 and themselves (and 1 is not prime!).

Factorize Reset

CALCULATOR RESULTS

What is prime factorization.

Prime factorization is when we split up a number into a product of its prime factors.

Every number (except prime numbers and the number 1) has its own unique set of prime factors.

To find out more about prime factorization, we hava dedicated page below.

  • What is Prime Factorization support page

How is Prime Factorization useful?

Prime factorization can be useful when working out the least common multiple, or the greatest common factor.

Prime factorization is also used in cryptography (the art of writing and solving codes) to encrypt or decrypt messages.

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

  • Greatest Common Factor Calculator

Our Greatest Common Factor calculator will tell you the highest common factor between 2 or more numbers.

It will also list the factors of each of the numbers and tell you whether they are coprime or not.

Greatest Common Factor Calculator Image

  • Least Common Multiple Calculator

Our Least Common Multiple Calculator will find the lowest common multiple of 2 or more numbers.

It will also show you the working out using a choice of two different methods.

There are also some worked examples.

least common multiple calculator image

Prime Factorization Worksheets

We have a collection of factor tree and prime factorization worksheets for students aged 6th grade and upwards.

Using factor trees is a great visual way of finding all the prime factors of a number.

We also have some problem solving, riddles and challenges on our Prime Factorization Worksheets page.

  • Factor Tree Worksheets (easier)
  • Prime Factorization Worksheets (harder)

Multiples and Factors Worksheets

These sheets are all about finding multiples and factors of different numbers.

They are a great way to introduce multiples and factors and to practice this skill.

  • Factors and Multiples Worksheet

Sieve of Erastosthenes

The Sieve of Erastosthenes is a method for finding what is a prime numbers between 2 and any given number.

Eratosthenes was a Greek mathematician (as well as being a poet, an astronomer and musician) who lived from about 276BC to 194BC.

If you want to find out more about his sieve for finding primes, and print out some Sieve of Eratosthenes worksheets, use the link below.

  • Sieve of Eratosthenes page

Want to find out more about primes?

Take a look at our Prime Number page which clearly describes what a prime numbers is and what they are not.

There are also many different questions about prime numbers answered, as well as information about the density of primes.

  • What is a Prime Number

Looking for more math calculators?

We have a wide range of free math calculators to help you.

Most of our calculators show you their working out so that you can see exactly what they have done to get the answer.

Our calculator hub page contains links to all of our calculators!

Free Fraction Calculator

  • Online Calculator Free

How to Print or Save these sheets 🖶

Need help with printing or saving? Follow these 3 steps to get your worksheets printed perfectly!

  • How to Print support

Return to Calculators Hub page

Return from Prime Factorization Calculator to Math Salamanders Homepage

Math-Salamanders.com

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.

New! Comments

TOP OF PAGE

Follow Me on Pinterest

© 2010-2024 Math Salamanders Limited. All Rights Reserved.

  • Privacy Policy
  • Copyright Policy

Prime Factorization

Prime numbers.

A Prime Number is:

a whole number above 1 that cannot be made by multiplying other whole numbers

The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19 and 23, and we have a prime number chart if you need more.

When it can  be made by multiplying other whole numbers it is a Composite Number .

"Factors" are the numbers you multiply together to get another number:

"Prime Factorization" is finding which prime numbers multiply together to make the original number.

Here are some examples:

Example: What are the prime factors of 12 ?

It is best to start working from the smallest prime number, which is 2, so let's check:

Yes, it divided exactly by 2. We have taken the first step!

But 6 is not a prime number, so we need to go further. Let's try 2 again:

Yes, that worked also. And 3 is a prime number, so we have the answer:

12 = 2 × 2 × 3

As you can see, every factor is a prime number , so the answer is right.

It is neater to show repeated numbers using  exponents :

  • Without exponents:  2 × 2 × 3
  • With exponents:  2 2 × 3

Example: What is the prime factorization of 147 ?

Can we divide 147 exactly by 2?

147 ÷ 2 = 73½

No we can't. The answer should be a whole number, and 73½ is not.

Let's try the next prime number, 3:

147 ÷ 3 = 49

That worked, now we try factoring 49.

The next prime, 5, does not work. But 7 does, so we get:

And that is as far as we need to go, because all the factors are prime numbers.

147 = 3 × 7 × 7 = 3 × 7 2

Example: What is the prime factorization of 17 ?

Hang on ... 17 is a Prime Number .

So that is as far as we can go.

Another Method

We just did factorization by starting at the smallest prime and working upwards.

But sometimes it is easier to break a number down into any factors we can ... then work those factor down to primes.

Example: What are the prime factors of 90 ?

Break 90 into 9 × 10

  • The prime factors of 9 are 3 and 3
  • The prime factors of 10 are 2 and 5

So the prime factors of 90 are 3, 3, 2 and 5

90 = 2 × 3 2 × 5

Factor Tree

And a "Factor Tree" can help: find any factors of the number, then the factors of those numbers, etc, until we can't factor any more.

Example: 48

48 = 8 × 6 , so we write down "8" and "6" below 48

Now we continue and factor 8 into 4 × 2

Then 4 into 2 × 2

And lastly 6 into 3 × 2

We can't factor any more, so we have found the prime factors.

Which reveals that 48 = 2 × 2 × 2 × 2 × 3

48 = 2 4 × 3

Why find Prime Factors?

A prime number can only be divided by 1 or itself, so it cannot be factored any further!

Every other whole number can be broken down into prime number factors.

This idea can be very useful when working with big numbers, such as in Cryptography.

Cryptography

Cryptography is the study of secret codes. Prime Factorization is very important to people who try to make (or break) secret codes based on numbers.

That is because factoring very large numbers is very hard, and can take computers a long time to do.

If you want to know more, the subject is "encryption" or "cryptography".

And here is another thing:

There is only one (unique!) set of prime factors for any number.

Example: the prime factors of 330 are 2, 3, 5 and 11

330 = 2 × 3 × 5 × 11

There is no other possible set of prime numbers that can be multiplied to make 330.

This idea is so important it is called the Fundamental Theorem of Arithmetic .

Prime Factorization Tool

OK, we have one more method ... use our Prime Factorization Tool that can work out the prime factors for numbers up to 4,294,967,296.

MatterofMath

Factors of 60 and How to Find Them

In this lesson, you are going to learn what the factors of 60 are, and the simple methods you can use to find these factors yourself.

Once you learn this, you will be able to use the same methods to find the factors of any number you want, which is a useful skill to have.

60 gets used a lot – every minute is 60 seconds, and every hour is 60 minutes! With more understanding of the numbers that make it up, you can use it more effectively in your life.

But first … what is a factor?

A factor is a whole number that divides another number perfectly, without leaving a remainder or a fractional part.

If you share sweets between people evenly, then the number of people and the number of sweets they each get are both factors of the number of sweets!

For example, if you have 60 sweets, you can share them fairly between 10 people. Each person would get 6 sweets, so 6 and 10 are both factors of 60.

60 sweets groups of 10

60’s Factors Pairs Primes Factorizing 60 How to Find 18’s Factors Prime Factorization Isn’t 18 Interesting? To Sum Up (Pun Intended!)

All the Factors of 60

Factors of 60

The factors of 60 are:

Because factors are related closely to division, you will sometimes hear them called divisors . Likewise, division is related to multiplication: a number is also a multiple of its factors.

6 and 10 are factors of 60, and 60 is a multiple of 6 and 10!

To make sure you’re familiar with both terms, both ‘factor’ and ‘divisor’ will be used throughout this article, even though they mean the same thing!

For a table of factors and multiples all the way up to 100 that you can use while you study, have a look here .

Factor Pairs of 60

When sharing the sweets out equally, the number of people and the number of sweets each person gets, are factors.

These two factors are called a factor pair , and they multiply together to make the original total number of sweets.

Here’s a list of all the factor pairs for 60:

Looking at all the numbers that appear in the factor pairs, you’ll see again that there are 12 factors of 60:

Sometimes 1 and 60 will be excluded from this list since every number has 1 and itself as a factor. The remaining factors are then called the proper factors:

Prime Factors of 60

The prime factors of a number are all the factors of a number that are also prime .

A prime number can only be divided by one, and itself.

For example, 3 can only be divided by 1 and 3, so it is prime.

1 itself isn’t a prime number. Here’s an article that explains why in a bit more detail.

From the list of factors of 60 above, there are 3 numbers that fit this description:

These 3 numbers are 60’s prime factors.

Factorizing 60

How to find the factors of 60.

So, now you know what all the factors of 60 are, but you also need to know how to find them.

Since factors come in factor pairs, once you find one factor, you can easily get another by finding the other half of the factor pair. You can find this other number of the pair by dividing the number you are factorizing by the factor you’ve already found.

For example, once you know that 5 is a factor of 60, you then work out 60÷5=12 , so 12 is another factor of 60 making the factor pair (5, 12) .

You will have found all the factors when you reach the whole number below the square root of 60.

60’s square root is 7.74… so you only need to check up to 7.

You can check whether a number is a factor using quick and easy tests you can perform in your head to check whether a number is divisible.

Remember, if a number can be divided then you have found a factor!

  • All whole numbers are divisible by 1, so there is nothing to check here – 1 is automatically a factor.
  • A number is divisibly by 2 if its last digit is a multiple of 2, so either 0, 2, 4, 6, or 8.
  • It is divisible by 3 if the sum of its digits is also divisible by 3.
  • 4 is a divisor if the last two digits form a number that is also divisible by four. If your number is only two digits long, you can use this trick instead: A number is divisible by 4 if half of the number is divisible by 2.
  • A number is divisible by 5 if its last digit is 5 or 0.
  • It can be divided by 6 if it is divisible by both 2 and 3. This is because 6=2×3 .
  • Remove the last digit and double it.
  • Then subtract the doubled digit from the rest of the number.
  • If the result is divisible by 7, then so was the original number.

… and for 60, you don’t need to go any further than this!

You can quickly see that 60 passes the tests for 1, 2, 3, 4, 5, and 6. The only one that might require a bit of work is 7, and it actually fails this one:

And 6 is not divisible by 7, so 60 isn’t divisible by 7.

Now we have found all the factors up to the square root of 60, so you can work out the other factors easily: divide 60 by each of the factors you’ve found to get the other factor in the pair.

  • 60 ÷ 1 = 60
  • 60 ÷ 2 = 30
  • 60 ÷ 3 = 20
  • 60 ÷ 4 = 15
  • 60 ÷ 5 = 12
  • 60 ÷ 6 = 10

That’s it! We’ve found all the factors!

Challenge – Try It Yourself!

These divisibility tests are incredibly powerful when it comes to checking the divisibility of large numbers. For instance…

  • It ends in 7, which is an odd number, meaning it’s not a multiple of 2, 4, or 6, and it’s also not a multiple of 5.
  • The sum of its digits is 6+3+7+7=23 which is not a multiple of 3, so it’s not a multiple of 3.

The only number of 2, 3, 4, 5, 6, and 7 which divides 6377 is 7.

Prime Factorization of 60

The first section established that the prime factors of 60 are 2, 3, and 5. Whole numbers have a special property: each one can be written as a product of its prime factors in one way and one way only.

This is called its prime factorization . The prime factorization of 60 looks like this:

You can find a number’s prime factorization using a factor tree . Here’s how they work:

  • Start by writing down the number you’re factorizing at the top
  • Split the number up into two factors, connected to the number above by ‘branches’.
  • Don’t choose 1 as a factor – it’s not going to be helpful!
  • If any of the factors are prime, circle them.
  • Repeat the process for any un-circled number: split numbers up into factor pairs and circle any primes until all the numbers left at the ends of branches have been circled, meaning they’re prime.
  • The circled primes that you’re left with are your prime factorization.

This is what you get for 60:

Prime tree creation for 60

  • First you just write down the number 60…
  • Then, choose any factor pair of 60, for example (4,15) , and split it up. Neither 4 nor 15 are prime, so don’t circle anything yet…
  • Split up 4 into 2 and 2 – it’s the only choice. 2 is prime, so get circling!
  • Likewise, split up 15 into 3 and 5.

Then, you’re left with

Of course, since these prime factorizations are unique, it shouldn’t matter how you go about doing the factor tree. You could have started with any factor pair of 60 at the first split and you’d end up with the same result.

If you don’t believe it, try it yourself!

Challenge – Try it Yourself!

a) 54 = 2 × 3 3

b) 130 = 2 × 5 × 13

c) 420 = 2 2 × 3 × 5 × 7

Here’s what your factor trees might look like.

Prime tree examples 420 54 130

If you chose different factors at the splitting-up stages, your trees will look different, but you should have the same answer if your math is correct!

Isn’t 60 Interesting

There are 60 seconds in a minute and 60 minutes in an hour: 60 is an important number!

Here are some facts about 60 that you might have seen before:

  • The time that it takes for a car to go from 0 to 60 miles per hour is often used to measure the performance of the car. 60mph is also a common speed limit used in lots of countries around the world.
  • People who have been married for 60 years celebrate their diamond anniversary. In 2012 Queen Elizabeth II celebrated her diamond jubilee, which marked her 60th year on the throne.
  • The highest score that a single dart can score in a game of darts is 60 – by landing on treble 20.
  • The length of a bowling lane in ten-pin bowling is exactly 60 feet.

It’s not surprising that the Ancient Babylonians decided to centre their number system around the number 60, since it has some interesting mathematical properties:

To Sum Up (Pun Intended!)

We factorized 60, and found the following 12 factors using divisibility tests:

We also found its prime factorization using a factor tree:

Make sure that you understand both of these methods.

If you’re still not totally sure, have a look at some other examples, like our articles on 18 and 52 , which walk you through factorizing some other numbers. You’ll find some interesting facts about those numbers there too!

If you have any solutions to the challenges, questions about the article, or your own fact about the number 60 that we missed here, let us know by leaving a comment below!

Leave a Comment Cancel Reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

  • 2.5 Prime Factorization and the Least Common Multiple
  • Introduction
  • 1.1 Introduction to Whole Numbers
  • 1.2 Add Whole Numbers
  • 1.3 Subtract Whole Numbers
  • 1.4 Multiply Whole Numbers
  • 1.5 Divide Whole Numbers
  • Key Concepts
  • Review Exercises
  • Practice Test
  • Introduction to the Language of Algebra
  • 2.1 Use the Language of Algebra
  • 2.2 Evaluate, Simplify, and Translate Expressions
  • 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
  • 2.4 Find Multiples and Factors
  • Introduction to Integers
  • 3.1 Introduction to Integers
  • 3.2 Add Integers
  • 3.3 Subtract Integers
  • 3.4 Multiply and Divide Integers
  • 3.5 Solve Equations Using Integers; The Division Property of Equality
  • Introduction to Fractions
  • 4.1 Visualize Fractions
  • 4.2 Multiply and Divide Fractions
  • 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
  • 4.4 Add and Subtract Fractions with Common Denominators
  • 4.5 Add and Subtract Fractions with Different Denominators
  • 4.6 Add and Subtract Mixed Numbers
  • 4.7 Solve Equations with Fractions
  • Introduction to Decimals
  • 5.1 Decimals
  • 5.2 Decimal Operations
  • 5.3 Decimals and Fractions
  • 5.4 Solve Equations with Decimals
  • 5.5 Averages and Probability
  • 5.6 Ratios and Rate
  • 5.7 Simplify and Use Square Roots
  • Introduction to Percents
  • 6.1 Understand Percent
  • 6.2 Solve General Applications of Percent
  • 6.3 Solve Sales Tax, Commission, and Discount Applications
  • 6.4 Solve Simple Interest Applications
  • 6.5 Solve Proportions and their Applications
  • Introduction to the Properties of Real Numbers
  • 7.1 Rational and Irrational Numbers
  • 7.2 Commutative and Associative Properties
  • 7.3 Distributive Property
  • 7.4 Properties of Identity, Inverses, and Zero
  • 7.5 Systems of Measurement
  • Introduction to Solving Linear Equations
  • 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
  • 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
  • 8.3 Solve Equations with Variables and Constants on Both Sides
  • 8.4 Solve Equations with Fraction or Decimal Coefficients
  • 9.1 Use a Problem Solving Strategy
  • 9.2 Solve Money Applications
  • 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
  • 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
  • 9.5 Solve Geometry Applications: Circles and Irregular Figures
  • 9.6 Solve Geometry Applications: Volume and Surface Area
  • 9.7 Solve a Formula for a Specific Variable
  • Introduction to Polynomials
  • 10.1 Add and Subtract Polynomials
  • 10.2 Use Multiplication Properties of Exponents
  • 10.3 Multiply Polynomials
  • 10.4 Divide Monomials
  • 10.5 Integer Exponents and Scientific Notation
  • 10.6 Introduction to Factoring Polynomials
  • 11.1 Use the Rectangular Coordinate System
  • 11.2 Graphing Linear Equations
  • 11.3 Graphing with Intercepts
  • 11.4 Understand Slope of a Line
  • A | Cumulative Review
  • B | Powers and Roots Tables
  • C | Geometric Formulas

Learning Objectives

By the end of this section, you will be able to:

  • Find the prime factorization of a composite number
  • Find the least common multiple (LCM) of two numbers

Be Prepared 2.12

Before you get started, take this readiness quiz.

Is 810 810 divisible by 2 , 3 , 5 , 6 , or 10 ? 2 , 3 , 5 , 6 , or 10 ? If you missed this problem, review Example 2.44 .

Be Prepared 2.13

Is 127 127 prime or composite? If you missed this problem, review Example 2.47 .

Write 2 ⋅ 2 ⋅ 2 ⋅ 2 2 ⋅ 2 ⋅ 2 ⋅ 2 in exponential notation. If you missed this problem, review Example 2.5 .

Find the Prime Factorization of a Composite Number

In the previous section, we found the factors of a number. Prime numbers have only two factors, the number 1 1 and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the prime factorization of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later in this course.

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number.

Manipulative Mathematics

You may want to refer to the following list of prime numbers less than 50 50 as you work through this section.

Prime Factorization Using the Factor Tree Method

One way to find the prime factorization of a number is to make a factor tree . We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree.

If a factor is prime, we circle it (like a bud on a tree), and do not factor that “branch” any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.

We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.

For example, let’s find the prime factorization of 36 . 36 . We can start with any factor pair such as 3 3 and 12 . 12 . We write 3 3 and 12 12 below 36 36 with branches connecting them.

The factor 3 3 is prime, so we circle it. The factor 12 12 is composite, so we need to find its factors. Let’s use 3 3 and 4 . 4 . We write these factors on the tree under the 12 . 12 .

The factor 3 3 is prime, so we circle it. The factor 4 4 is composite, and it factors into 2 · 2 . 2 · 2 . We write these factors under the 4 . 4 . Since 2 2 is prime, we circle both 2 s . 2 s .

The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.

In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.

Note that we could have started our factor tree with any factor pair of 36 . 36 . We chose 12 12 and 3 , 3 , but the same result would have been the same if we had started with 2 2 and 18 , 4 18 , 4 and 9 , or 6 and 6 . 9 , or 6 and 6 .

Find the prime factorization of a composite number using the tree method.

  • Step 1. Find any factor pair of the given number, and use these numbers to create two branches.
  • Step 2. If a factor is prime, that branch is complete. Circle the prime.
  • Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.
  • Step 4. Write the composite number as the product of all the circled primes.

Example 2.48

Find the prime factorization of 48 48 using the factor tree method.

Check this on your own by multiplying all the factors together. The result should be 48 . 48 .

Try It 2.95

Find the prime factorization using the factor tree method: 80 80

Try It 2.96

Find the prime factorization using the factor tree method: 60 60

Example 2.49

Find the prime factorization of 84 using the factor tree method.

Draw a factor tree of 84 . 84 .

Try It 2.97

Find the prime factorization using the factor tree method: 126 126

Try It 2.98

Find the prime factorization using the factor tree method: 294 294

Prime Factorization Using the Ladder Method

The ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.

To begin building the “ladder,” divide the given number by its smallest prime factor. For example, to start the ladder for 36 , 36 , we divide 36 36 by 2 , 2 , the smallest prime factor of 36 . 36 .

To add a “step” to the ladder, we continue dividing by the same prime until it no longer divides evenly.

Then we divide by the next prime; so we divide 9 9 by 3 . 3 .

We continue dividing up the ladder in this way until the quotient is prime. Since the quotient, 3 , 3 , is prime, we stop here.

Do you see why the ladder method is sometimes called stacked division?

The prime factorization is the product of all the primes on the sides and top of the ladder.

Notice that the result is the same as we obtained with the factor tree method.

Find the prime factorization of a composite number using the ladder method.

  • Step 1. Divide the number by the smallest prime.
  • Step 2. Continue dividing by that prime until it no longer divides evenly.
  • Step 3. Divide by the next prime until it no longer divides evenly.
  • Step 4. Continue until the quotient is a prime.
  • Step 5. Write the composite number as the product of all the primes on the sides and top of the ladder.

Example 2.50

Find the prime factorization of 120 120 using the ladder method.

Check this yourself by multiplying the factors. The result should be 120 . 120 .

Try It 2.99

Find the prime factorization using the ladder method: 80 80

Try It 2.100

Find the prime factorization using the ladder method: 60 60

Example 2.51

Find the prime factorization of 48 48 using the ladder method.

Try It 2.101

Find the prime factorization using the ladder method. 126 126

Try It 2.102

Find the prime factorization using the ladder method. 294 294

Find the Least Common Multiple (LCM) of Two Numbers

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

Listing Multiples Method

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of 10 10 and 25 . 25 . We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.

We see that 50 50 and 100 100 appear in both lists. They are common multiples of 10 10 and 25 . 25 . We would find more common multiples if we continued the list of multiples for each.

The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of 10 10 and 25 25 is 50 . 50 .

Find the least common multiple (LCM) of two numbers by listing multiples.

  • Step 1. List the first several multiples of each number.
  • Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
  • Step 3. Look for the smallest number that is common to both lists.
  • Step 4. This number is the LCM.

Example 2.52

Find the LCM of 15 15 and 20 20 by listing multiples.

List the first several multiples of 15 15 and of 20 . 20 . Identify the first common multiple.

15: 15 , 30 , 45 , 60 , 75 , 90 , 105 , 120 20: 20 , 40 , 60 , 80 , 100 , 120 , 140 , 160 15: 15 , 30 , 45 , 60 , 75 , 90 , 105 , 120 20: 20 , 40 , 60 , 80 , 100 , 120 , 140 , 160

The smallest number to appear on both lists is 60 , 60 , so 60 60 is the least common multiple of 15 15 and 20 . 20 .

Notice that 120 120 is on both lists, too. It is a common multiple, but it is not the least common multiple.

Try It 2.103

Find the least common multiple (LCM) of the given numbers: 9 and 12 9 and 12

Try It 2.104

Find the least common multiple (LCM) of the given numbers: 18 and 24 18 and 24

Prime Factors Method

Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of 12 12 and 18 . 18 .

We start by finding the prime factorization of each number.

Then we write each number as a product of primes, matching primes vertically when possible.

Now we bring down the primes in each column. The LCM is the product of these factors.

Notice that the prime factors of 12 12 and the prime factors of 18 18 are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that 36 36 is the least common multiple.

Find the LCM using the prime factors method.

  • Step 1. Find the prime factorization of each number.
  • Step 2. Write each number as a product of primes, matching primes vertically when possible.
  • Step 3. Bring down the primes in each column.
  • Step 4. Multiply the factors to get the LCM.

Example 2.53

Find the LCM of 15 15 and 18 18 using the prime factors method.

Try It 2.105

Find the LCM using the prime factors method. 15 and 20 15 and 20

Try It 2.106

Find the LCM using the prime factors method. 15 and 35 15 and 35

Example 2.54

Find the LCM of 50 50 and 100 100 using the prime factors method.

Try It 2.107

Find the LCM using the prime factors method: 55 , 88 55 , 88

Try It 2.108

Find the LCM using the prime factors method: 60 , 72 60 , 72

ACCESS ADDITIONAL ONLINE RESOURCES

  • Ex 1: Prime Factorization
  • Ex 2: Prime Factorization
  • Ex 3: Prime Factorization
  • Ex 1: Prime Factorization Using Stacked Division
  • Ex 2: Prime Factorization Using Stacked Division
  • The Least Common Multiple
  • Example: Determining the Least Common Multiple Using a List of Multiples
  • Example: Determining the Least Common Multiple Using Prime Factorization

Section 2.5 Exercises

Practice makes perfect.

In the following exercises, find the prime factorization of each number using the factor tree method.

In the following exercises, find the prime factorization of each number using the ladder method.

In the following exercises, find the prime factorization of each number using any method.

In the following exercises, find the least common multiple (LCM) by listing multiples.

8 , 12 8 , 12

4 , 3 4 , 3

6 , 15 6 , 15

12 , 16 12 , 16

30 , 40 30 , 40

20 , 30 20 , 30

60 , 75 60 , 75

44 , 55 44 , 55

In the following exercises, find the least common multiple (LCM) by using the prime factors method.

24 , 30 24 , 30

28 , 40 28 , 40

70 , 84 70 , 84

84 , 90 84 , 90

In the following exercises, find the least common multiple (LCM) using any method.

6 , 21 6 , 21

9 , 15 9 , 15

32 , 40 32 , 40

Everyday Math

Grocery shopping Hot dogs are sold in packages of ten, but hot dog buns come in packs of eight. What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? (Hint: it is the LCM!)

Grocery shopping Paper plates are sold in packages of 12 12 and party cups come in packs of 8 . 8 . What is the smallest number of plates and cups you can purchase if you want to have the same number of each? (Hint: it is the LCM!)

Writing Exercises

Do you prefer to find the prime factorization of a composite number by using the factor tree method or the ladder method? Why?

Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction
  • Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
  • Publisher/website: OpenStax
  • Book title: Prealgebra 2e
  • Publication date: Mar 11, 2020
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/prealgebra-2e/pages/1-introduction
  • Section URL: https://openstax.org/books/prealgebra-2e/pages/2-5-prime-factorization-and-the-least-common-multiple

© Jan 23, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

  • Math Article

Prime Factorization

Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers. Here, all the concepts of prime factors and prime factorization methods have been explained which will help the students understand how to find the prime factors of a number easily.

The simplest algorithm to find the prime factors of a number is to keep on dividing the original number by prime factors until we get the remainder equal to 1. For example, prime factorizing the number 30 we get, 30/2 = 15, 15/3 = 5, 5/5 = 1. Since we received the remainder, it cannot be further factorized. Therefore, 30 = 2 x 3 x 5, where 2,3 and 5 are prime factors.

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19 and so on. These prime numbers when multiplied with any natural numbers produce composite numbers.

In this article, let us discuss the definition of prime factorization, different methods to find the prime factors of a number with solved examples.

What is Prime Factorization?

Prime factorization is defined as a way of finding the prime factors of a number, such that the original number is evenly divisible by these factors. As we know, a composite number has more than two factors, therefore, this method is applicable only for composite numbers and not for prime numbers.

For example, the prime factors of 126 will be 2, 3 and 7 as 2 × 3 × 3 × 7 = 126 and 2, 3, 7 are prime numbers.

Prime factorization Examples

  • Prime factorization of 12 is 2 × 2 × 3 = 2 2  × 3
  • Prime factorization of 18 is 2 × 3 × 3 = 2 × 3 2
  • Prime factorization of 24 is 2 × 2 × 2 × 3 = 2 3  × 3
  • Prime factorization of 20 is 2 × 2 × 5 = 2 2  × 5
  • Prime factorization of 36 is 2 × 2 × 3 × 3 = 2² × 3²
  • Prime Factorization of HCF and LCM

The prime numbers when multiplied by any natural numbers or whole numbers (but not 0), gives composite numbers. So basically prime factorization is performed on the composite numbers to factorize them and find the prime factors. This method is also used in the case of finding the HCF (Highest Common Factor) and LCM (Least Common Multiple) of any given set of numbers.

If any two numbers are given, then the highest common factor is the largest factor present in both the numbers whereas the least common multiple is the smallest common multiple of both the numbers.

Prime Factors of a Number

Prime factors of a number are the set of prime numbers which when multiplied by together give the actual number. Also, we can say, the prime factors divide the number completely. It is similar to factoring a number and considering only the prime numbers among the factors. For example, the prime factors of 6 will be 2 and 3, the prime factors of 26 will be 13 and 2, etc.

Prime Factorization Methods

The most commonly used prime factorization methods are:

Division Method

Factor tree method.

The steps to calculate the prime factors of a number is similar to the process of finding the factors of a large number. Follow the below steps to find the prime factors of a number using the division method:

  • Step 1: Divide the given number by the smallest prime number. In this case, the smallest prime number should divide the number exactly.
  • Step 2: Again, divide the quotient by the smallest prime number.
  • Step 3: Repeat the process, until the quotient becomes 1.
  • Step 4: Finally, multiply all the prime factors

Example of Division Method for Prime Factorization:

Below is a detailed step-by-step process of prime factorization by taking 460 as an example.

  • Step 1: Divide 460 by the least prime number i.e. 2.

          So, 460 ÷ 2 = 230

  • Step 2: Again Divide 230 with the least prime number (which is again 2).

          Now, 160 ÷ 2 = 115

  • Step 3: Divide again with the least prime number which will be 5.

          So, 115 ÷ 5 = 23

  • Step 4: As 23 is a prime number, divide it with itself to get 1.

          Now, the prime factors of 460 will be 2 2 x 5 x 23

To find the prime factorization of the given number using factor tree method, follow the below steps:

  • Step 1: Consider the given number as the root of the tree
  • Step 2: Write down the pair of factors as the branches of a tree
  • Step 3: Again factorize the composite factors, and write down the factors pairs as the branches
  • Step 4: Repeat the step, until to find the prime factors of all the composite factors

In factor tree, the factors of a number are found and then those numbers are further factorized until we reach the closure. Suppose we have to find the factors of 60 and 282 using a factor tree. Then see the diagram given below to understand the concept.

Prime factorisation Factor Tree Method

In the above figure, we can number 60 is first factorized into two numbers i.e. 6 and 10. Again, 6 and 10 is factorized to get the prime factors of 6 and 10, such that;

and 10 = 2 x 5

If we write the prime factors of 60 altogether, then;

Prime factorization of 60 = 6 x 10 = 2 x 3 x 2 x 5

Same is the case for number 282, such as;

282 = 2 x 141 = 2 x 3 x 47

So in both cases, a tree structure is formed.

Related Articles

  • Prime numbers
  • Factorisation
  • Square Root By Prime Factorization

Prime Factorization Solved Examples

An example question is given below which will help to understand the process of calculating the prime factors of a number easily.

Q.1:  Find the prime factors of 1240.

∴ The Prime Factors of 1240 will be 2 3 × 5 × 31.

Q.2: Find the prime factors of 544.

Therefore, the prime factors of 544 are 2 5 x 17.

Prime Factorization Worksheet (Questions)

  • What is the prime factorization of 48?

Write the prime factors of 2664 without using exponents.

  • Is 40 = 20 × 2 an example of prime factorization process? Justify.
  • Write 6393 as a product of prime factors.

Frequently Asked Questions on Prime Factorization

Define prime factorization..

Prime factorization is the process of finding the prime numbers, which are multiplied together to get the original number. For example, the prime factors of 16 are 2 × 2 × 2 × 2. This can also be written as 2 4

What are the two different methods to find the prime factors of a number?

The two different methods to find the prime factors of a number are: Division method Factor tree method

Write down the prime factorization of 13.

The prime factorization of 13 is 13. Because the prime factors of 13 are 1 and 13. As 1, and 13 are prime numbers, the prime factorization of 13 is written as 1×13, which is equal to 13.

What is the prime factorization of 999?

The prime factorization of 999 can be easily found using the factor tree method. The prime factorization of 999 is 3 3 ×37 1 , which is equal to 3×3×3×37. The numbers 3 and 37 are the prime numbers.

Find out the prime factors of 15.

The prime factors of 15 are 3×5. When the prime numbers 3 and 15 are multiplied together, we get the original number 15.

Learn more about interesting maths concepts by registering at BYJU’S – The Learning App. Also, get various maths-related materials for exams like sample papers, question papers, and additional important notes and questions.

Quiz Image

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

Visit BYJU’S for all Maths related queries and study materials

Your result is as below

Request OTP on Voice Call

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Post My Comment

how to write 60 as a product of prime factors

the prime factorization of 99999

3*2 x 41*1 x 271*1

What are prime factors

Factors of a number that are prime numbers are called prime factors. These prime factors if multiplied together gives the original number. For example, prime factors of 15 is 3 and 5. 3 x 5 = 15

What is co prime number

Two or more numbers are said to be co-prime numbers, if the only common factor between them is 1. Learn more about coprime numbers with examples at BYJU’S.

what are the prime factors of 36

Prime Factorisation of 36 = 2 x 2 x 3 x 3

The prime factors of 2664 are: 2 x 2 x 2 x 3 x 3 x 37

Prime factorisation method (a)42025

Prime factorisation method for 42025: 42025 ÷ 5 = 8405 8405 ÷ 5 = 1681 1681 ÷ 41 = 41 41 ÷ 41 = 1 Thus, prime factorisation of 42025 = 5 × 5 × 41 × 41

how to write 60 as a product of prime factors

  • Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

close

  • CORE CURRICULUM
  • LITERACY > CORE CURRICULUM > Into Literature, 6-12" data-element-type="header nav submenu" title="Into Literature, 6-12" aria-label="Into Literature, 6-12"> Into Literature, 6-12
  • LITERACY > CORE CURRICULUM > Into Reading, K-6" data-element-type="header nav submenu" title="Into Reading, K-6" aria-label="Into Reading, K-6"> Into Reading, K-6
  • INTERVENTION
  • LITERACY > INTERVENTION > English 3D, 4-12" data-element-type="header nav submenu" title="English 3D, 4-12" aria-label="English 3D, 4-12"> English 3D, 4-12
  • LITERACY > INTERVENTION > Read 180, 3-12" data-element-type="header nav submenu" title="Read 180, 3-12" aria-label="Read 180, 3-12"> Read 180, 3-12
  • LITERACY > READERS > Hero Academy Leveled Libraries, PreK-4" data-element-type="header nav submenu" title="Hero Academy Leveled Libraries, PreK-4" aria-label="Hero Academy Leveled Libraries, PreK-4"> Hero Academy Leveled Libraries, PreK-4
  • LITERACY > READERS > HMH Reads Digital Library, K-5" data-element-type="header nav submenu" title="HMH Reads Digital Library, K-5" aria-label="HMH Reads Digital Library, K-5"> HMH Reads Digital Library, K-5
  • LITERACY > READERS > inFact Leveled Libraries, K-5" data-element-type="header nav submenu" title="inFact Leveled Libraries, K-5" aria-label="inFact Leveled Libraries, K-5"> inFact Leveled Libraries, K-5
  • LITERACY > READERS > Rigby PM, K-5" data-element-type="header nav submenu" title="Rigby PM, K-5" aria-label="Rigby PM, K-5"> Rigby PM, K-5
  • LITERACY > READERS > Science & Engineering Leveled Readers, K-5" data-element-type="header nav submenu" title="Science & Engineering Leveled Readers, K-5" aria-label="Science & Engineering Leveled Readers, K-5"> Science & Engineering Leveled Readers, K-5
  • SUPPLEMENTAL
  • LITERACY > SUPPLEMENTAL > A Chance in the World SEL, 8-12" data-element-type="header nav submenu" title="A Chance in the World SEL, 8-12" aria-label="A Chance in the World SEL, 8-12"> A Chance in the World SEL, 8-12
  • LITERACY > SUPPLEMENTAL > Amira Learning, K-6" data-element-type="header nav submenu" title="Amira Learning, K-6" aria-label="Amira Learning, K-6"> Amira Learning, K-6
  • LITERACY > SUPPLEMENTAL > Classcraft, K-8" data-element-type="header nav submenu" title="Classcraft, K-8" aria-label="Classcraft, K-8"> Classcraft, K-8
  • LITERACY > SUPPLEMENTAL > JillE Literacy, K-3" data-element-type="header nav submenu" title="JillE Literacy, K-3" aria-label="JillE Literacy, K-3"> JillE Literacy, K-3
  • LITERACY > SUPPLEMENTAL > Waggle, K-8" data-element-type="header nav submenu" title="Waggle, K-8" aria-label="Waggle, K-8"> Waggle, K-8
  • LITERACY > SUPPLEMENTAL > Writable, 3-12" data-element-type="header nav submenu" title="Writable, 3-12" aria-label="Writable, 3-12"> Writable, 3-12
  • LITERACY > SUPPLEMENTAL > ASSESSMENT" data-element-type="header nav submenu" title="ASSESSMENT" aria-label="ASSESSMENT"> ASSESSMENT
  • MATH > CORE CURRICULUM > Arriba las Matematicas, K-8" data-element-type="header nav submenu" title="Arriba las Matematicas, K-8" aria-label="Arriba las Matematicas, K-8"> Arriba las Matematicas, K-8
  • MATH > CORE CURRICULUM > Go Math!, K-6" data-element-type="header nav submenu" title="Go Math!, K-6" aria-label="Go Math!, K-6"> Go Math!, K-6
  • MATH > CORE CURRICULUM > Into Algebra 1, Geometry, Algebra 2, 8-12" data-element-type="header nav submenu" title="Into Algebra 1, Geometry, Algebra 2, 8-12" aria-label="Into Algebra 1, Geometry, Algebra 2, 8-12"> Into Algebra 1, Geometry, Algebra 2, 8-12
  • MATH > CORE CURRICULUM > Into Math, K-8" data-element-type="header nav submenu" title="Into Math, K-8" aria-label="Into Math, K-8"> Into Math, K-8
  • MATH > CORE CURRICULUM > Math Expressions, PreK-6" data-element-type="header nav submenu" title="Math Expressions, PreK-6" aria-label="Math Expressions, PreK-6"> Math Expressions, PreK-6
  • MATH > CORE CURRICULUM > Math in Focus, K-8" data-element-type="header nav submenu" title="Math in Focus, K-8" aria-label="Math in Focus, K-8"> Math in Focus, K-8
  • MATH > SUPPLEMENTAL > Classcraft, K-8" data-element-type="header nav submenu" title="Classcraft, K-8" aria-label="Classcraft, K-8"> Classcraft, K-8
  • MATH > SUPPLEMENTAL > Waggle, K-8" data-element-type="header nav submenu" title="Waggle, K-8" aria-label="Waggle, K-8"> Waggle, K-8
  • MATH > INTERVENTION > Math 180, 5-12" data-element-type="header nav submenu" title="Math 180, 5-12" aria-label="Math 180, 5-12"> Math 180, 5-12
  • SCIENCE > CORE CURRICULUM > Into Science, K-5" data-element-type="header nav submenu" title="Into Science, K-5" aria-label="Into Science, K-5"> Into Science, K-5
  • SCIENCE > CORE CURRICULUM > Into Science, 6-8" data-element-type="header nav submenu" title="Into Science, 6-8" aria-label="Into Science, 6-8"> Into Science, 6-8
  • SCIENCE > CORE CURRICULUM > Science Dimensions, K-12" data-element-type="header nav submenu" title="Science Dimensions, K-12" aria-label="Science Dimensions, K-12"> Science Dimensions, K-12
  • SCIENCE > READERS > inFact Leveled Readers, K-5" data-element-type="header nav submenu" title="inFact Leveled Readers, K-5" aria-label="inFact Leveled Readers, K-5"> inFact Leveled Readers, K-5
  • SCIENCE > READERS > Science & Engineering Leveled Readers, K-5" data-element-type="header nav submenu" title="Science & Engineering Leveled Readers, K-5" aria-label="Science & Engineering Leveled Readers, K-5"> Science & Engineering Leveled Readers, K-5
  • SCIENCE > READERS > ScienceSaurus, K-8" data-element-type="header nav submenu" title="ScienceSaurus, K-8" aria-label="ScienceSaurus, K-8"> ScienceSaurus, K-8
  • SOCIAL STUDIES > CORE CURRICULUM > HMH Social Studies, 6-12" data-element-type="header nav submenu" title="HMH Social Studies, 6-12" aria-label="HMH Social Studies, 6-12"> HMH Social Studies, 6-12
  • SOCIAL STUDIES > SUPPLEMENTAL > Writable" data-element-type="header nav submenu" title="Writable" aria-label="Writable"> Writable
  • For Teachers
  • PROFESSIONAL DEVELOPMENT > For Teachers > Coachly" data-element-type="header nav submenu" title="Coachly" aria-label="Coachly"> Coachly
  • PROFESSIONAL DEVELOPMENT > For Teachers > Teacher's Corner" data-element-type="header nav submenu" title="Teacher's Corner" aria-label="Teacher's Corner"> Teacher's Corner
  • PROFESSIONAL DEVELOPMENT > For Teachers > Live Online Courses" data-element-type="header nav submenu" title="Live Online Courses" aria-label="Live Online Courses"> Live Online Courses
  • For Leaders
  • PROFESSIONAL DEVELOPMENT > For Leaders > The Center for Model Schools (formerly ICLE)" data-element-type="header nav submenu" title="The Center for Model Schools (formerly ICLE)" aria-label="The Center for Model Schools (formerly ICLE)"> The Center for Model Schools (formerly ICLE)
  • MORE > undefined > Assessment" data-element-type="header nav submenu" title="Assessment" aria-label="Assessment"> Assessment
  • MORE > undefined > Early Learning" data-element-type="header nav submenu" title="Early Learning" aria-label="Early Learning"> Early Learning
  • MORE > undefined > English Language Development" data-element-type="header nav submenu" title="English Language Development" aria-label="English Language Development"> English Language Development
  • MORE > undefined > Homeschool" data-element-type="header nav submenu" title="Homeschool" aria-label="Homeschool"> Homeschool
  • MORE > undefined > Intervention" data-element-type="header nav submenu" title="Intervention" aria-label="Intervention"> Intervention
  • MORE > undefined > Literacy" data-element-type="header nav submenu" title="Literacy" aria-label="Literacy"> Literacy
  • MORE > undefined > Mathematics" data-element-type="header nav submenu" title="Mathematics" aria-label="Mathematics"> Mathematics
  • MORE > undefined > Professional Development" data-element-type="header nav submenu" title="Professional Development" aria-label="Professional Development"> Professional Development
  • MORE > undefined > Science" data-element-type="header nav submenu" title="Science" aria-label="Science"> Science
  • MORE > undefined > undefined" data-element-type="header nav submenu">
  • MORE > undefined > Social and Emotional Learning" data-element-type="header nav submenu" title="Social and Emotional Learning" aria-label="Social and Emotional Learning"> Social and Emotional Learning
  • MORE > undefined > Social Studies" data-element-type="header nav submenu" title="Social Studies" aria-label="Social Studies"> Social Studies
  • MORE > undefined > Special Education" data-element-type="header nav submenu" title="Special Education" aria-label="Special Education"> Special Education
  • MORE > undefined > Summer School" data-element-type="header nav submenu" title="Summer School" aria-label="Summer School"> Summer School
  • BROWSE RESOURCES
  • BROWSE RESOURCES > Classroom Activities" data-element-type="header nav submenu" title="Classroom Activities" aria-label="Classroom Activities"> Classroom Activities
  • BROWSE RESOURCES > Customer Success Stories" data-element-type="header nav submenu" title="Customer Success Stories" aria-label="Customer Success Stories"> Customer Success Stories
  • BROWSE RESOURCES > Digital Samples" data-element-type="header nav submenu" title="Digital Samples" aria-label="Digital Samples"> Digital Samples
  • BROWSE RESOURCES > Events & Webinars" data-element-type="header nav submenu" title="Events & Webinars" aria-label="Events & Webinars"> Events & Webinars
  • BROWSE RESOURCES > Grants & Funding" data-element-type="header nav submenu" title="Grants & Funding" aria-label="Grants & Funding"> Grants & Funding
  • BROWSE RESOURCES > International" data-element-type="header nav submenu" title="International" aria-label="International"> International
  • BROWSE RESOURCES > Research Library" data-element-type="header nav submenu" title="Research Library" aria-label="Research Library"> Research Library
  • BROWSE RESOURCES > Shaped - HMH Blog" data-element-type="header nav submenu" title="Shaped - HMH Blog" aria-label="Shaped - HMH Blog"> Shaped - HMH Blog
  • CUSTOMER SUPPORT
  • CUSTOMER SUPPORT > Contact Sales" data-element-type="header nav submenu" title="Contact Sales" aria-label="Contact Sales"> Contact Sales
  • CUSTOMER SUPPORT > Customer Service & Technical Support Portal" data-element-type="header nav submenu" title="Customer Service & Technical Support Portal" aria-label="Customer Service & Technical Support Portal"> Customer Service & Technical Support Portal
  • CUSTOMER SUPPORT > Platform Login" data-element-type="header nav submenu" title="Platform Login" aria-label="Platform Login"> Platform Login
  • Learn about us
  • Learn about us > About" data-element-type="header nav submenu" title="About" aria-label="About"> About
  • Learn about us > Diversity, Equity, and Inclusion" data-element-type="header nav submenu" title="Diversity, Equity, and Inclusion" aria-label="Diversity, Equity, and Inclusion"> Diversity, Equity, and Inclusion
  • Learn about us > Environmental, Social, and Governance" data-element-type="header nav submenu" title="Environmental, Social, and Governance" aria-label="Environmental, Social, and Governance"> Environmental, Social, and Governance
  • Learn about us > News Announcements" data-element-type="header nav submenu" title="News Announcements" aria-label="News Announcements"> News Announcements
  • Learn about us > Our Legacy" data-element-type="header nav submenu" title="Our Legacy" aria-label="Our Legacy"> Our Legacy
  • Learn about us > Social Responsibility" data-element-type="header nav submenu" title="Social Responsibility" aria-label="Social Responsibility"> Social Responsibility
  • Learn about us > Supplier Diversity" data-element-type="header nav submenu" title="Supplier Diversity" aria-label="Supplier Diversity"> Supplier Diversity
  • Join Us > Careers" data-element-type="header nav submenu" title="Careers" aria-label="Careers"> Careers
  • Join Us > Educator Input Panel" data-element-type="header nav submenu" title="Educator Input Panel" aria-label="Educator Input Panel"> Educator Input Panel
  • Join Us > Suppliers and Vendors" data-element-type="header nav submenu" title="Suppliers and Vendors" aria-label="Suppliers and Vendors"> Suppliers and Vendors
  • Divisions > Center for Model Schools (formerly ICLE)" data-element-type="header nav submenu" title="Center for Model Schools (formerly ICLE)" aria-label="Center for Model Schools (formerly ICLE)"> Center for Model Schools (formerly ICLE)
  • Divisions > Heinemann" data-element-type="header nav submenu" title="Heinemann" aria-label="Heinemann"> Heinemann
  • Divisions > NWEA" data-element-type="header nav submenu" title="NWEA" aria-label="NWEA"> NWEA
  • Platform Login

SOCIAL STUDIES

PROFESSIONAL DEVELOPMENT

Teaching Product of Prime Factors

Richard Blankman Blog 800X800 3

Number theory , or the study of integers (the counting numbers 1, 2, 3..., their opposites –1, –2, –3..., and 0), has fascinated mathematicians for years. Prime numbers , a concept introduced to most students in Grades 4 and up, are fundamental to number theory. They form the basic building blocks for all integers.

A prime number is a counting number that only has two factors, itself and one. Counting numbers which have more than two factors (such as 6, whose factors are 1, 2, 3, and 6), are said to be composite numbers . The number 1 only has one factor and usually isn't considered either prime or composite.

  • Key standard: Determine whether a given number is prime or composite, and find all factors for a whole number. (Grade 4)

Why Do Prime Factors Matter?

It's the age-old question that math teachers everywhere must contend with. When will I use this? One notable example is with cryptography , or the study of creating and deciphering codes. With the help of a computer, it is easy to multiply two prime numbers. However, it can be extremely difficult to factor a number. Because of this, when a website sends and receives information securely—something especially important for financial or medical websites, for example—you can bet there are prime numbers behind the scenes. Prime numbers also show up in a variety of surprising contexts, including physics, music, and even in the arrival of cicadas !

There is another place where prime numbers show up often, and it's easy to overlook when discussing applications: math! The study of pure mathematics is a topic that people practice, study, and share without worrying about where else it might apply, similar to how a musician does not need to ask how music applies to the real world. Number theory is an extremely rich topic that is central to college courses, research papers, and other branches of mathematics. Mathematicians of all stripes no doubt encounter number theory many times along their academic and professional journeys.

how to write 60 as a product of prime factors

Writing a Product of Prime Factors

When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, we can write the number 72 as a product of prime factors: \(72=2^3 \cdot 3^2\). The expression \(2^3 \cdot 3^2\) is said to be the prime factorization of 72. The Fundamental Theorem of Arithmetic states that every composite number can be factored uniquely (except for the order of the factors) into a product of prime factors. What this means is that how you choose to factor a number into prime factors makes no difference. When you are done, the prime factorizations are essentially the same.

Examine the two factor trees for 72 shown below.

New factor tree

When we get done factoring using either set of factors to start with, we still have three factors of 2 and two factors of 3, or \(2^3 \cdot 3^2\). This would be true if we had started to factor 72 as 24 times 3, 4 times 18, or any other pair of factors for 72.

Knowing rules for divisibility is helpful when factoring a number. For example, if a whole number ends in 0, 2, 4, 6, or 8, we could always start the factoring process by dividing by 2. It should be noted that because 2 only has two factors, 1 and 2, it is the only even prime number.

Another way to factor a number other than using factor trees is to start dividing by prime numbers:

Teaching product prime factors inline 2 division expression

Once again, we can see that \(72=2^3 \cdot 3^2\).

Also key to writing the prime factorization of a number is an understanding of exponents . An exponent tells how many times the base is used as a factor. In the prime factorization of \(72=2^3 \cdot 3^2\), the 2 is used as a factor three times and the 3 is used as a factor twice.

There is a strategy we can use to figure out whether a number is prime. Find the square root (with the help of a calculator if needed), and only check prime numbers less than or equal to it. For example, to see if 131 is prime, because the square root is between 11 and 12, we only need to check for divisibility by 2, 3, 5, 7, and 11. There is no need to check 13, since 13 2 = 169, which is greater than 131. This works because if a prime number greater than 13 divided 131, then the other factor would have to be less than 13—which we're already checking!

Introducing the Concept: Finding Prime Factors

Making sure your students' work is neat and orderly will help prevent them from losing factors when constructing factor trees. Have them check their prime factorizations by multiplying the factors to see if they get the original number.

Prerequisite Skills and Concepts: Students will need to know and be able to use exponents. They also will find it helpful to know the rules of divisibility for 2, 3, 4, 5, 9 and 10.

Write the number 48 on the board.

  • Ask : Who can give me two numbers whose product is 48? Students should identify pairs of numbers like 6 and 8, 4 and 12, or 3 and 16. Take one of the pairs of factors and create a factor tree for the prime factorization of 48 where all students can see it.

Teaching product prime factors inline 3 factor tree

  • Ask : How many factors of two are there? (4) How do I express that using an exponent? Students should say to write it as \(2^4\). If they don't, remind them that the exponent tells how many times the base is taken as a factor. Finish writing the prime factorization on the board as \(2^4 \cdot 3\). Next, find the prime factorization for 48 using a different set of factors.
  • Ask: What do you notice about the prime factorization of 48 for this set of factors? Students should notice that the prime factorization of 48 is \(2^4 \cdot 3\) for both of them.
  • Say : There is a theorem in mathematics that says when we factor a number into a product of prime numbers, it can only be done one way, not counting the order of the factors. Illustrate this concept by showing them that the prime factorization of 48 could also be written as \(3 \cdot 2^4\), but mathematically, that's the same thing as \(2^4 \cdot 3\).
  • Say : Now let's try one on your own. Find the prime factorization of 60 by creating a factor tree for 60. Have all students independently factor 60. As they complete their factorizations, observe what students do and take note of different approaches and visual representations. Ask for a student volunteer to factor 60 for the entire class to see.
  • Ask : Who factored 60 differently? Have students who factored 60 differently (either by starting with different factors or by visually representing the factor tree differently) show their work to the class. Ask students to describe similarities and differences in the factorizations. If no one used different factors, show the class a factorization that starts with a different set of factors for 60 and have students identify similarities and differences between your factor tree and other students'.
  • Ask : If I said the prime factorization of 36 is 2 2 • 9, would I be right? The students should say no, because 9 is not a prime number. If they don't, remind them that the prime factorization of a number means all the factors must be prime and 9 is not a prime number.

Place the following composite numbers on the board and ask them to write the prime factorization for each one using factor trees: 24, 56, 63, and 46.

Developing the Concept: Product of Prime Numbers

Now that students can find the prime factorization for numbers which are familiar products, it is time for them to use their rules for divisibility and other notions to find the prime factorization of unfamiliar numbers. Write the number 91 on the board.

  • Say : Yesterday, we wrote some numbers in their prime factorization form.
  • Ask : Who can write 91 as a product of prime numbers? Many students might say it can't be done, because they will recognize that 2, 3, 4, 5, 9 and 10 don't divide it. They may not try to see if 7 divides it, which it does. If they don't recognize that 7 divides 91, demonstrate it for them. The prime factorization of 91 is \(7 \cdot 13\). Next, write the number 240 on the board.
  • Ask : Who can tell me two numbers whose product is 240? Students are likely to say 10 and 24. If not, ask them to use their rules for divisibility to see if they can find two numbers. Create a factor tree for 240 like the one below.

Teaching product prime factors inline 4 factor tree

  • Ask : How many factors of two are there in the prime factorization of 240? (4) Who can tell me how to write the prime factorization of 240? (2 4 • 3 • 5) Facilitate a discussion around different ways to factor 240 and the pros and cons of each method. If you start with 2 and 120, you end up with the same prime factorization in the end, but you end up with a "one-sided tree" that some students may find more difficult to work with. Have students identify ways that they prefer to factor and guide them to explain their reasoning .
  • Say : Since the prime factorization of 240 is 2 4 • 3 • 5, the only prime numbers which divide this number are 2, 3 and 5. Prime numbers like 7 and 11 will not divide the number, because they do not appear in the prime factorization of the number. Write the number 180 on the board.
  • Ask : What two numbers might we start with to find the prime factorization of 180? What other numbers could we use? Encourage students to find a variety of pairs, such as 10 and 18 or 9 and 20. If no one mentions either pair, suggest them both as possibilities. Have half the students use 10 and 18 and the other half use 9 and 20. Have two students create the two factors for the class to see.
  • Ask : If the prime factorization of a number is 2 2 • 5 • 7, what can you tell me about the number?
  • Ask : If the prime factorization of a number is 3 3 • 11, what can you tell me about this number? Repeat the previous exercise with a new number. Some possible observations: Because \(3^2\) is a factor, the number is divisible by 9 and the sum of the number's digits is a multiple of nine. Because the product of odd numbers is always odd, the number is an odd number. They might also tell you that it is a composite number, five is not a factor of the number, and so on. Give them the following numbers and ask them to find their prime factorization: 231, 117, and 175. Also give the following prime factorizations of numbers and ask them to write down at least two things they know about both the number represented: \(3^2 \cdot 5^2\), \(2^3 \cdot 3 \cdot 13\), and \(2^2 \cdot 3 \cdot 5\). You can of course adjust both the numbers and factorizations to match what your students are ready for.

Wrap-Up and Assessment Hints

Finding the prime factorization of numbers will strengthen your students' basic facts and understanding of multiplication. Students who do not know their basic multiplication facts will likely struggle with this, because they do not recognize products such as 24 or 63 readily. Turning the problem around and giving them the prime factorization of a number and asking them what they know about the number without multiplying it out is a good way to assess their understanding of the divisibility rules, the concept of factoring, and multiplication in general.

To develop students' conceptual understanding and help them grow into procedurally fluent mathematicians, explore HMH Into Math , our core solution for K–8 math instruction.

  • Activities & Lessons

Related Reading

Reading Intervention Elementary Hero

Research-Based Reading Interventions for Elementary Students

Shaped Contributor

February 13, 2024

Teaching Writing to Elementary Students Hero

How to Teach Writing to Elementary Students without Fear

Kristen Eannetta Instructional Coach

February 8, 2024

Presidents Day Lesson Plans And Activities

9 Presidents' Day Lesson Plans and Activities to Foster Engaged Citizens

Brenda Iasevoli Shaped Executive Editor

February 1, 2024

Factoring Calculator

Enter the expression you want to factor in the editor.

Difference of Squares : a 2   –   b 2   =   ( a   +   b ) ( a   –   b )

Please ensure that your password is at least 8 characters and contains each of the following:

  • a special character: @$#!%*?&

IMAGES

  1. 60 as a Product of Prime Factors in Index Form

    how to write 60 as a product of prime factors

  2. Factors of 60

    how to write 60 as a product of prime factors

  3. What Is Prime Factor

    how to write 60 as a product of prime factors

  4. Write the prime factorization of 60

    how to write 60 as a product of prime factors

  5. factor trees and prime factorization

    how to write 60 as a product of prime factors

  6. How to Write a Number as a Product of Its Prime Factors

    how to write 60 as a product of prime factors

VIDEO

  1. Prime Factors of 124 #shorts

  2. Prime factors of 30

  3. Prime Factors of 72 #shorts

  4. Prime Factors of 75 #shorts

  5. Prime Factors of 49 and 50

  6. Write the prime factors of the following numbers

COMMENTS

  1. Factors of 60

    The prime factorization of 60 can be done by multiplying all its prime factors such that the product is 60. Let us learn about all factors of 60, the prime factorization of 60, and the factor tree of 60 in this article. ... forms the first pair, (2, 30) forms the second pair and the list goes on as shown. So, as we write 1 as the factor of 60 ...

  2. Factors of 60? How to Find the Factors of 60 by Prime Factorization

    Example 3: Find the product of all the prime factors of 60. Solution: Prime factors of 60 are 2, 3, and 5. Product of prime factors = 2 × 3 × 5 = 30. So, the product of all the prime factors of 60 is 30. Example 4: A school library has a collection of 60 books on the history of America.

  3. Finding prime factors

    Sometimes you might be asked to write a number as the product close product The answer when two or more numbers are multiplied together, eg the product of 4 and 15 is 60. of its prime factors.

  4. Prime Factorization Calculator

    Prime numbers are widely used in number theory due to the fundamental theorem of arithmetic. This theorem states that natural numbers greater than 1 are either prime, or can be factored as a product of prime numbers. As an example, the number 60 can be factored into a product of prime numbers as follows: 60 = 5 × 3 × 2 × 2

  5. Prime factors of 60

    Prime Factorization of 60 it is expressing 60 as the product of prime factors. In other words it is finding which prime numbers should be multiplied together to make 60. ... Now we have all the Prime Factors for number 60. It is: 2, 2, 3, 5. Or you may also write it in exponential form: 2 2 × 3 × 5. Prime Factor Tree of 60.

  6. Factors of 60

    The steps to find the factors for 60 are given below: Step 1: First, write the number 60 in your notebook. Step 2: Find the two numbers, which on multiplication give 60, say 1 and 60, such as 1 × 60 = 60. Step 3: Let us write another pair of numbers that gives 60 on multiplication, such as 2 and 30, i.e. 2 × 30 = 60.

  7. Prime Factors of 60

    Here is the math to illustrate: 60 ÷ 2 = 30. 30 ÷ 2 = 15. 15 ÷ 3 = 5. 5 ÷ 5 = 1. Again, all the prime numbers you used to divide above are the Prime Factors of 60. Thus, the Prime Factors of 60 are: 2, 2, 3, 5.

  8. Prime Factorization Calculator

    Prime factorization is when we break a number down into factors that are only prime numbers. If we look at the above example with 20, the factors are 1, 2, 4, 5, 10, 20. The best place to start is to find at least one initial factor that is prime. Since 5 is prime, we can start with 4 * 5. Notice that 4 is not prime, so we break 4 down into 2 * 2.

  9. Prime factorization (video)

    This video explains the concept of prime numbers and how to find the prime factorization of a number using a factorization tree. It also shows how to write the prime factorization using exponential notation. A prime number is a number that is only divisible by itself and one. Created by Sal Khan and Monterey Institute for Technology and Education.

  10. Prime Factorization

    Prime factorization is the process of writing a number as the product of prime numbers.Prime numbers are the numbers that have only two factors, 1 and the number itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, and so on are prime numbers. Prime factorization of any number means to represent that number as a product of prime numbers. For example, the prime factorization of 40 can be done in ...

  11. Factors of 60

    Step 1: Start with 60. Step 2: Now find the smallest prime factor that divides 60. It's 2 as shown in figure for first branch under 60. Step 3: Continue dividing each branch by prime factors until you reach only prime numbers. Step 4: The prime factors are now at the ends of each branch.

  12. Prime Factorization Calculator

    This calculator presents: For the first 5000 prime numbers, this calculator indicates the index of the prime number. The nth prime number is denoted as Prime [n], so Prime [1] = 2, Prime [2] = 3, Prime [3] = 5, and so on. The limit on the input number to factor is less than 10,000,000,000,000 (less than 10 trillion or a maximum of 13 digits).

  13. Lesson Explainer: Prime Factorization with Exponents

    To find the prime factorization of 60, start by writing 60 in the first step (if it helps, you can think of each step like upside-down division). Then, write any prime factor of 60 next to it and the quotient underneath in a new step. So, we can choose 2 as a prime factor of 60, and then the quotient to write in the next step is 6 0 ÷ 2 = 3 0.

  14. 60 as a Product of Prime Factors

    Learn From Our Maths Experts! In This Article we Talk About 60 as a Product of Prime Factors. Get in Touch for Help Learning Maths.

  15. Express 60 as a product of its prime factors.

    We divide 60 by 2 as many times as possible to get 60 = 15 x 2 x 2. 2 is a prime number so we don't need to break the 2's down any more. Instead we break 15 down. 15 doesn't divide by 2 so we try the next prime number: 3 Divide 15 by 3 to get 15 = 5 x 3, and so 60 = 5 x 3 x 2 x 2. Now we collect the factors, so our final answer is 60 = 5 x 3 x 2 2.

  16. Prime Factorization Calculator

    Prime Factorization Calculator - How to use it. Input any number values into our Prime Factorization Calculator and it will find quickly find all of the factors and then re-write the number as a unique product of its prime factors. It will also show you what the number looks like in exponential form. Note: prime numbers cannot be written as a ...

  17. Prime Factorization

    Let's try 2 again: 6 ÷ 2 = 3. Yes, that worked also. And 3 is a prime number, so we have the answer: 12 = 2 × 2 × 3. As you can see, every factor is a prime number, so the answer is right. It is neater to show repeated numbers using exponents: Without exponents: 2 × 2 × 3. With exponents: 22 × 3.

  18. Factors of 60 and How to Find Them · Matter of Math

    Prime Factorization of 60. The first section established that the prime factors of 60 are 2, 3, and 5. Whole numbers have a special property: each one can be written as a product of its prime factors in one way and one way only. This is called its prime factorization. The prime factorization of 60 looks like this:

  19. How to Write a Number as a Product of Its Prime Factors

    Example: Write 140 as the Product of Its Prime Factors. The first step is to find two numbers that multiply together to make 140. There are multiple pairs that will do this, e.g., 2 × 70, 4 × 35, 5 × 28, etc., but it doesn't matter which one we choose. We will find out more about the reason why later on. In this example, let's start with 10 ...

  20. 2.5 Prime Factorization and the Least Common Multiple

    Step 1. Find any factor pair of the given number, and use these numbers to create two branches. Step 2. If a factor is prime, that branch is complete. Circle the prime. Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process. Step 4.

  21. Prime Factorization

    In the above figure, we can number 60 is first factorized into two numbers i.e. 6 and 10. Again, 6 and 10 is factorized to get the prime factors of 6 and 10, such that; 6 = 2 x 3. and 10 = 2 x 5. If we write the prime factors of 60 altogether, then; Prime factorization of 60 = 6 x 10 = 2 x 3 x 2 x 5. Same is the case for number 282, such as;

  22. Teaching Product of Prime Factors

    When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, we can write the number 72 as a product of prime factors: 72 = 2 3 ⋅ 3 2. The expression 2 3 ⋅ 3 2 is said to be the prime factorization of 72. The Fundamental Theorem of Arithmetic states that every ...

  23. Factoring Calculator

    Factoring Calculator. Enter the expression you want to factor in the editor. The Factoring Calculator transforms complex expressions into a product of simpler factors. It can factor expressions with polynomials involving any number of vaiables as well as more complex functions. Difference of Squares: a2 - b2 = (a + b)(a - b) a 2 - b 2 ...