Factors, prime factors and multiples are topics that regularly appear in GCSE maths questions, but they can be surprisingly easy to muddle up.

The words sound similar, and when you're under pressure in an exam it can be difficult to remember whether you're supposed to divide, multiply or draw a factor tree.

So in the video accompanying this article, I go through a simple way of remembering the difference.

What is a factor?

A factor is a whole number that divides exactly into another number.

For example, the factors of 12 are:

1, 2, 3, 4, 6 and 12

That's because each of these numbers divides into 12 without leaving a remainder.

One way of thinking about factors is to ask:

“What numbers can I multiply together to make my original number?”

For 12:

1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

So 1, 2, 3, 4, 6 and 12 are all factors of 12.

What are prime factors?

A prime number is a number greater than 1 that has exactly two factors: 1 and itself.

Prime factors are therefore the prime numbers that multiply together to make another number.

This is where factor trees are really useful.

Suppose we want to find the prime factors of 16.

Start with 16 and split it into two factors. We could choose:

16 = 4 × 4

Neither 4 is prime, so we split them again:

4 = 2 × 2

That leaves us with:

2 × 2 × 2 × 2

Two is a prime number, so there is nowhere else for our factor tree to go.

We can then write our final answer using index notation:

16 = 2⁴

because we're multiplying four 2s together.

A simple way to remember prime factors

If you find yourself forgetting which method to use, try drawing a little picture of a tree in your notes and writing:

PRIME FACTORS

inside it.

It's a simple visual reminder:

Prime factors → factor tree 🌳

Sometimes a small visual association is much easier to remember in an exam than another definition you've tried to learn by heart.

So what is a multiple?

This is where students can sometimes mix things up.

With factors, we're looking for numbers that go into our original number.

With multiples, we're going in the other direction.

A multiple is what we get when we multiply a number by a whole number.

Take 5, for example:

5 × 1 = 5
5 × 2 = 10
5 × 3 = 15
5 × 4 = 20
5 × 5 = 25

So:

5, 10, 15, 20 and 25 are all multiples of 5.

And the list keeps going.

A useful way to think about multiples is simply as the number's times table.

Factor or multiple?

Here's the distinction I'd try to remember:

Factors go INTO a number.
Multiples come FROM multiplying a number.

For example, using 20:

5 is a factor of 20 because 20 ÷ 5 = 4.

20 is a multiple of 5 because 5 × 4 = 20.

It's the relationship between the numbers that matters.

Try these yourself

Before looking at the answers, have a go at these:

1. List all the factors of 18.

2. Write 24 as a product of its prime factors.

3. Write down the first five multiples of 7.

4. Is 6 a factor or a multiple of 24?

5. Is 24 a factor or a multiple of 6?

Answers

1. 1, 2, 3, 6, 9, 18

2. 2³ × 3

3. 7, 14, 21, 28, 35

4. 6 is a factor of 24.

5. 24 is a multiple of 6.

Don't just memorise it — find a way to remember it

One of the things we regularly do at Clara James is look for a way of making something memorable for the individual child.

For one student, writing the definitions repeatedly might work perfectly.

For somebody else, a little tree doodle might be the thing they picture when the question appears in their GCSE exam.

Neither is inherently better.

The important thing is finding the way that helps you remember.

If you're revising factors, prime factors and multiples for GCSE maths, have a watch of the video above and then try creating your own factor trees and lists of multiples.

And if there's another GCSE maths topic you'd like me to break down simply, let me know. It might become the next video.