Factors and Multiples

Two words that work together

Factors and multiples are two of the most useful ideas in maths, and they are closely linked. The good news is that if you know your Times Tables, you already know most of this — you just need the right words for it.

Let's meet them one at a time.

What is a factor?

A factor is a whole number that divides exactly into another number, leaving no remainder. You can think of factors as the numbers that "fit perfectly" inside a number.

Take 12. Which numbers divide into it exactly?

• 12 ÷ 1 = 12 ✔
• 12 ÷ 2 = 6 ✔
• 12 ÷ 3 = 4 ✔
• 12 ÷ 4 = 3 ✔
• 12 ÷ 6 = 2 ✔
• 12 ÷ 12 = 1 ✔

So the factors of 12 are 1, 2, 3, 4, 6 and 12. Notice that 5 is not a factor, because 12 ÷ 5 leaves a remainder.

Factors come in pairs

Here is a neat idea that helps you find every factor without missing any: factors come in pairs that multiply to make the number. This is the why behind the pairing — each division has a matching multiplication.

For 12, the factor pairs are:

Factor pair	Product
1 × 12	12
2 × 6	12
3 × 4	12

To find all factors, start at 1 and work upwards, writing each pair, until the two numbers in the pair meet or cross over. This way you never forget a factor.

What is a multiple?

A multiple is what you get when you multiply a number by a whole number (1, 2, 3, and so on). The multiples of a number are simply its times table.

The multiples of 4 are:

4, 8, 12, 16, 20, 24, 28, ...

Each one is 4 multiplied by a whole number: 4×1, 4×2, 4×3, and so on. Unlike factors, the list of multiples never ends — you can always make a bigger one.

Factors vs multiples: the key difference

Students often mix these up, so here is the clearest way to remember:

• A factor is smaller than or equal to the number — it fits inside it. (3 is a factor of 12.)
• A multiple is bigger than or equal to the number — it is built from it. (12 is a multiple of 3.)

In fact, "3 is a factor of 12" and "12 is a multiple of 3" describe the same relationship from two directions.

Common factors

When two numbers share a factor, it is called a common factor.

Example — common factors of 8 and 12.

1. Factors of 8: 1, 2, 4, 8.
2. Factors of 12: 1, 2, 3, 4, 6, 12.
3. The numbers in both lists are 1, 2 and 4.
4. So the common factors are 1, 2 and 4, and the highest common factor (HCF) is 4.

The highest common factor is exactly what you use to simplify fractions. To reduce 8/12, you divide top and bottom by their HCF of 4, giving 2/3.

Common multiples

When two numbers share a multiple, it is a common multiple.

Example — common multiples of 3 and 4.

1. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...
2. Multiples of 4: 4, 8, 12, 16, 20, 24, ...
3. Numbers in both lists: 12, 24, 36, ...
4. The smallest one is 12, called the lowest common multiple (LCM).

The lowest common multiple is exactly what you use to find a common denominator when adding fractions with different denominators.

A practice activity

Try this "factor rainbow" game:

1. Pick the number 24. Write 1 at the far left and 24 at the far right.
2. Work inwards in pairs: 1–24, 2–12, 3–8, 4–6. Draw a curved line connecting each pair like a rainbow.
3. You should find the factors 1, 2, 3, 4, 6, 8, 12, 24 — eight factors in all.

Then answer these:

• (a) List all factors of 18.
• (b) Write the first five multiples of 6.
• (c) Find the highest common factor of 6 and 9.
• (d) Find the lowest common multiple of 2 and 5.

Answers: (a) 1, 2, 3, 6, 9, 18; (b) 6, 12, 18, 24, 30; (c) 3; (d) 10.

Where this leads

Factors and multiples are the foundation for Prime Numbers, simplifying fractions, and adding fractions with different denominators. The more fluent your times tables become, the faster you will spot factors and multiples. Keep practising, and these ideas will make a huge range of later maths feel easy.