Prime factors

Every number can be written as a product of prime numbers.

(Remember: ‘product’ means ‘times’ or ‘multiply’.)

For example 40 = 2 x 2 x 2 x 5

126 = 2 x 3 x 3 x 7

28 = 2 x 2 x 7

How do we write a number as a product of its prime factors?

Worked example

Write 24 as a product of its prime factors.

There are lots of ways to solve this. Here are two:

Solution 1

Start with the smallest prime number that divides into 24, in this case 2. (2 divides into all even numbers.) So we can write 24 as:

24 = 2 x 12

Now think of the smallest prime number that divides into 12. Again we can use 2, and write the 12 as 2 x 6, to give.

24 = 2 x 2 x 6

6 also divides by 2 (6 = 2 x 3), so we have:

24 = 2 x 2 x 2 x 3

2, 2, 2 and 3 are all prime numbers, so we have our answer.

In short, we would write the solution as:

24 = 2 x 12

= 2 x 2 x 6

= 2 x 2 x 2 x 3

Solution 2

We can also use a factor tree:

This shows us that 24 = 2 x 2 x 2 x 3

Remember that we can write these factors in index form. In this case

24 = 2 x 2 x 2 x 3, so

24 = 2³ x 3