Factoring numbers

Last updated: 2 Feb 2007

Factors

The factors of a particular number are all the numbers which divide it evenly.

For example the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30 because they all divide 30 evenly.

30 = 1 x 30 = 2 x 15 = 3 x 10 = 5 x 6

Another definition of a prime number is a positive number that has exactly two factors (itself and one).

Prime factors

Every number can be uniquely expressed as the product of its prime factors.

For example, we have already seen that the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. Therefore its prime factors are 2, 3 and 5.

2 x 3 x 5 = 30

Sometimes the same prime factor is repeated in the product. For example if we take 12. Its factors are 1, 2, 3, 4, 6 and 12, therefore it's prime factors are 2 and 3.

`2 xx 2 xx 3 = 2^2 xx 3 = 12`

How to find all the prime factors of a number?

To find find all the prime factors of a number you just keep dividing it by prime numbers until you are left with a prime. It is easiest if you start with 2, the lowest prime, and work up.

For example let us find the prime factors of 84 so that we can express 84 as a product of those factors.

First divide by 2

`84 -: 2 = 42` - Prime factor of 2

Then divide by 2 again

`42 -: 2 = 21` - Another prime factor of 2

21 is not divisible by 2 so divide by 3

`21 -: 3 = 7` - Prime factor of 3.

We are left with 7 which is a prime and so is the final prime factor of 84.

Therefore

`84 = 2^2 xx 3 xx 7`

Next page: Divisibility tests