How to Find Factors and Multiples | KS3 Maths

A factor of a number divides into it exactly. The quickest way to see them all is to build rectangles: take 24 tiles and every rectangle you can make from them — 4 by 6, 3 by 8, 2 by 12, 1 by 24 — hands you two factors at once. That is why factors always come in pairs, and it is how you know when your list is finished.

What is a factor?

A factor of a number is a whole number that divides into it exactly, leaving nothing over.

The picture worth holding is a rectangle of tiles. If you can arrange 24 tiles into 4 rows of 6, then $4 \times 6 = 24$, and both 4 and 6 divide into 24 exactly. Both are factors. Because a rectangle has two side lengths, every rectangle you find hands you two factors at once — which is the reason factors come in pairs, and the reason you can stop hunting once the rectangles run out.

This is the same tile picture used in square and cube numbers, where the rectangle happens to be a perfect square.

How do you find all the factors of a number?

Work through the rectangles in order: 1 row, 2 rows, 3 rows, and so on. Each one that comes out even gives you a pair.

Worked example

List all the factors of 24

Start with 24 tiles and try each number of rows in turn.

1 row of 24 — $1 \times 24 = 24$. Factors: 1 and 24.

2 rows of 12 — $2 \times 12 = 24$. Factors: 2 and 12.

3 rows of 8 — $3 \times 8 = 24$. Factors: 3 and 8.

4 rows of 6 — $4 \times 6 = 24$. Factors: 4 and 6.

5 rows? 24 does not split into 5 equal rows, so 5 is not a factor.

6 rows of 4 — we have already had this pair the other way round. Once the pairs start repeating, you are done.

$\text{factors of } 24 = 1,\ 2,\ 3,\ 4,\ 6,\ 8,\ 12,\ 24$

Eight factors, from four rectangles.

What is a multiple?

A multiple is what you land on counting up in a number. The multiples of 6 are $6, 12, 18, 24, 30, \dots$ — they carry on forever, so you are usually asked for the first few.

Worked example

Write the first five multiples of 6

Start at 6 and keep adding 6. In tiles: start with one block of six, then add another block, and another.

$6$    $6 + 6 = 12$    $12 + 6 = 18$    $18 + 6 = 24$    $24 + 6 = 30$

$6,\ 12,\ 18,\ 24,\ 30$

They are just the 6 times table, which is why a multiple of 6 is sometimes described as "a number in the 6 times table".

How are factors and multiples related?

They are the same fact read in two directions, which is exactly why they get muddled.

Look at the fourth multiple of 6. It is 24 — the number from the first example. So:

• 6 is a factor of 24, because 6 fits into 24 exactly.
• 24 is a multiple of 6, because 24 is what you reach counting up in sixes.

One statement, two viewpoints. A factor is smaller than the number (or equal to it); a multiple is bigger (or equal). If you can remember which way round the picture goes, you never have to remember the words.

How do you know when you have found every factor?

You stop when the pairs start repeating — and there is an exact moment when that happens.

Worked example

List all the factors of 36

Work through the rows as before.

$1 \times 36$  ·  $2 \times 18$  ·  $3 \times 12$  ·  $4 \times 9$

5 rows? 36 does not split into 5 equal rows, so 5 is not a factor.

6 rows of 6 — and here is something new. This rectangle is a square: 6 pairs with itself.

That square is the stopping point. The next rectangle would be $9 \times 4$, which is $4 \times 9$ turned on its side — one we have already had. So is $12 \times 3$, and $18 \times 2$, and $36 \times 1$.

$\text{factors of } 36 = 1,\ 2,\ 3,\ 4,\ 6,\ 9,\ 12,\ 18,\ 36$

Nine factors — an odd number. Every other number on this page had an even count, because the factors came in pairs. 36 has an odd count precisely because one of its "pairs" is a double. That is true of every square number and only of square numbers.

Going further

This is why you only test up to the square root. The stopping point above is not a coincidence about 36 — it happens for every number. Keep going while your test number multiplied by itself is still below the number, and stop as soon as it goes past. To check whether 97 has any factors you only need to go up to 9, because $9 \times 9 = 81$ is still below 97 while $10 \times 10 = 100$ is already past it — and any factor bigger than 9 would need a partner smaller than 9 that you have already tried. That shortcut is what makes testing large numbers for primes practical, and it is the heart of the prime numbers lesson.

Every number has at least two factors — except 1. Every number $n$ has $1$ and $n$ in its list. The number 1 is the exception: its two "ends" are the same number, so it has exactly one factor.

Where you'll use this next

Factors and multiples are the machinery behind a lot of what comes after. Finding the highest common factor and the lowest common multiple is exactly this skill applied to two numbers at once. Simplifying a fraction means dividing top and bottom by a common factor. Adding fractions means finding a common multiple of the denominators — which is why the adding and subtracting fractions method starts by listing multiples.