Factorising Numbers

Factorisation is a complicated sounding word that simply means finding the individual numbers that multiply together to give a larger number. These individual numbers are called factors. For example, 6 and 8 are factors of 48 because 6 x 8 = 48.

However, we can sometimes factorise these factors, as is the case with both 6 and 8. If we cannot factorise any further we have found the prime factors. The prime factors are prime numbers, since they won't have factors themselves. (A prime number is a number which is only divisible by itself and 1.)

This can probably be most easily seen with a few examples. Say we have a number 6. We note that it is an even number, so we know that 2 will be one of the factors.

6 = 2 x 3

Both 2 and 3 are prime numbers, so we can't factorise 6 any further. Thus we can say that the prime factors of 6 are 2 and 3.

Let's have a crack at a bigger number, say 10,080. A straight forward way to do this is to nibble away at the number by repeatedly taking out small factors. For the first few factors, each time we take out a factor of 2 the paired factor is still even, so we know there is another factor of 2.

At this point we know that there aren't any more 2s, since we're down to an odd number. It's pretty obvious that another factor is 5, so let's do that.

And with that we're down to a number that we should recognise from our times tables.

That last line was just rearranging the factors to put the 3s in a tidier place. So the prime factors of 10,080 are 2, 2, 2, 2, 2, 3, 3, 5, and 7. This is a slightly long-winded way of writing the factors, so if we want we can use exponents.

10,080 = 2⁵ x 3² x 5 x 7

As far as the end result is concerned it doesn't matter at all what order we find them in, although if we can break the number (or any of its later factors) apart into two big factors early on it gets the factors within range of our times tables earlier. This makes factorising quickly and more efficient because the factors become easier to break apart quickly. For example, take out a factor of 10, and then factorise it and the other factor at the same time. In 1008, 100 and 8 are both divisible by 4, then a bit later in 126, 12 and 6 are both divisible by 6. This means we can more easily break those numbers up because we don't get remainders carrying over into other columns.

That took six lines of working instead of ten and we weren't even trying very hard.

There are some very handy Factorising Tricks we can use to make it easier to find factors, although sometimes those tricks will not be much use.

And that's as far as we can go, because 3,607 and 3,803 are prime numbers. So the prime factors of 1,234,567,890 are 2, 3, 3, 5, 3607 and 3803.