The product of prime factors made easy. Prime factor decomposition.

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Before you can work out the product of prime factors first you will need to know what prime numbers and factors are. A prime number is a number that can only be divided by 1 and itself. For example, 13 is prime because it only can be divided by 1 and 13. Factors are all the numbers that divide exactly into another number without a remainder. For example, the factors of 12 are 1,2,3,4,6 and 12. Also the prime factors of 12 are 2 and 3 (factors and prime)

To find the product of prime factors of a number divide your original number by a prime factor. Then divide the resulting number by another prime factor. Keep repeating this process until you can’t divide anymore (until your left with a prime number). Then your answer will be all your prime factors multiplied together (as product means multiply).

Try remembering your first 10 prime numbers 2,3,5,7,11,13,17,19,23 and 29 as these will be the numbers that you are dividing your number by.

Example 1

Write 80 as a product of prime factors:

80 ÷ 2 = 40 (divide by 2)

40 ÷ 2 = 20

20 ÷ 2 = 10

10 ÷ 2 = 5

So our product of prime factors of 80 is 2 × 2 × 2 × 2 × 5. Make sure you include your finishing number in your product of prime factors.

Example 2

Write 210 as a product of prime factors.

210 ÷ 2 = 105

105 ÷ 5 = 21

21 ÷ 3 = 7

So our answer is 2 × 3 × 5 × 7 (put your prime factors in size order)

Example 3

Write 64 as a product of prime factors.

64 ÷ 2 = 32

32 ÷ 2 = 16

16 ÷ 2 = 8

8 ÷ 2 = 4

4 ÷ 2 = 2

So 64 as a product of prime factors can be written as 2 × 2 × 2 × 2 × 2 × 2.

You can also write this answer using powers as 2⁶ (since there are six 2’s are being multiplied together).

Extra Tips

Multiply your product of prime factors together to make sure it gives you the original number. If it works then you know your answer is correct.

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