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Level 4Grade 4skill: factors_multiples· 8 min read

Factors and Multiples Explained

In short: A factor divides a number exactly; a multiple is that number times a whole number. Factors fit inside a number and come in pairs, so you only need to test up to the middle. Multiples are just the times-table, counting up forever.

How to find factors and multiples

Two words that sound alike but pull in opposite directions. A factor of a number divides into it exactly, with nothing left over. A multiple is what you get when you multiply that number by a whole number. So the factors of 6 are the small numbers hidden inside it — 1, 2, 3, 6 — while the multiples of 6 are the big numbers it builds up to — 6, 12, 18, 24, and on forever.

The quickest way to keep them straight: factors fit into a number, multiples march past it. If you already know your times tables to 12, you know almost everything you need — you're just reading them in a different direction. To find factors, test whole numbers in order and keep the ones that divide cleanly. Better still, write each one next to its partner, because factors always come two at a time.

  1. Decide what the question wants: factors (numbers that divide it exactly) or multiples (its times-table).
  2. For multiples, count up in equal steps: n, then 2n, 3n, 4n, and so on.
  3. For factors, test 1, 2, 3, 4... in turn, keeping any that divide with no remainder.
  4. Write each factor beside its pair, for example for 6 write 2 beside 3.
  5. Stop testing once the pairs begin to meet in the middle — you already have them all.

Why factor pairs are the whole trick Why it works

Factor pairs
Every factor has exactly one partner it multiplies with

24 = 1x24, 2x12, 3x8, 4x6 — four pairs give all eight factors

Multiples never stop
The multiples of n are n, 2n, 3n... going on without end

Multiples of 9: 9, 18, 27, 36, 45 — you could keep going all day

1 and itself
1 and the number itself are always factors

For 15, that gives 1 and 15; the other pair is 3 and 5

Stop at the middle
When the pairs start repeating, you have found every factor

For 36, the pair 6x6 is the middle, so there is nothing left to check past 6

Here is the one idea that turns factor-hunting from guesswork into a method: factors come in pairs, and the two numbers in each pair close in on each other. 1 pairs with the biggest, 2 with the next biggest, and so on until they meet in the middle — like 6x6 for 36. The moment they meet, you already hold every factor. You never have to check past the middle, which is why finding all the factors of 36 takes only six little checks, not thirty-six.

Worked examples

Find all the factors of 18. = 1, 2, 3, 6, 9, 18 (six factors)
  1. Start at 1: 1 x 18 = 18, so 1 and 18 are both factors.
  2. Try 2: 18 / 2 = 9 exactly, so 2 and 9 are factors.
  3. Try 3: 18 / 3 = 6 exactly, so 3 and 6 are factors.
  4. Try 4: 18 / 4 leaves a remainder, so 4 is not a factor. 5 fails too.
  5. The next number, 6, has already appeared as 3's partner — the pairs have met, so stop.
List the first five multiples of 7. = 7, 14, 21, 28, 35
  1. Multiply 7 by 1, 2, 3, 4, 5 in turn.
  2. 7x1 = 7, 7x2 = 14, 7x3 = 21, 7x4 = 28, 7x5 = 35.
  3. These are the 7-times table read straight off — no dividing needed.
Is 8 a factor of 30? = No. 8 is not a factor of 30.
  1. A factor must divide with no remainder.
  2. Try 30 / 8: 8 x 3 = 24 and 8 x 4 = 32, so 30 lands between them.
  3. 30 / 8 = 3 remainder 6 — it does not divide exactly.
Find the common factors of 12 and 18. = 1, 2, 3 and 6 are the common factors
  1. Factors of 12: 1, 2, 3, 4, 6, 12.
  2. Factors of 18: 1, 2, 3, 6, 9, 18.
  3. Circle the numbers that appear in both lists.

Four slips to watch for

Giving a multiple when the question asked for a factor

✗ 10

Asked for a factor of 5, this student multiplied instead of divided and answered 10 (that's 5x2, a multiple). Factors are smaller than or equal to the number; multiples are bigger than or equal to it — 5 is both its own largest factor and its own first multiple.

Fix: Ask: does it divide into 5 exactly? 10 cannot fit inside 5, so it is a multiple, not a factor. The factors of 5 are just 1 and 5.

wrong_operation

Listing a non-factor from a slightly-wrong fact

✗ 6

Hunting for factors of 56, this student wrote 6, half-remembering 6x9 = 56. But 6x9 = 54, not 56 — the neighbouring fact tricked them. The pair that really makes 56 is 7x8.

Fix: Check it: 56 / 6 = 9 remainder 2, so 6 does not divide evenly. Trust the division, not a shaky recall of the table.

table_neighbour

Dropping a zero in a multiple

✗ 12

Asked for the 6th multiple of 20, this student computed 6x2 = 12 and stopped. They multiplied by 2 instead of 20, losing the tens place value.

Fix: 20 ends in a zero, so 6 x 20 = 120. Multiply the digits, then put the zero back: 6x2 = 12, then 120.

place_value_slip

Missing the middle factor

✗ 4

Counting the factors of 16, this student listed 1, 2, 8, 16 and answered 4 factors. They skipped the middle factor 4, whose partner is itself (4x4 = 16), which is easy to overlook.

Fix: Write the pairs: 1x16, 2x8, 4x4. A square number like 16 has a factor that pairs with itself, so 16 has 5 factors, not 4.

off_by_one

Practice: factors and multiples

Work through these one at a time. Find factors by pairing up and stopping at the middle; find multiples by counting up in equal steps. If you slip, follow the link to the exact mistake and see how to fix it.

Keep going

First, make sure of

Frequently asked questions

What is the difference between a factor and a multiple?

A factor divides a number exactly with no remainder, so factors are smaller than or equal to the number (the factors of 6 are 1, 2, 3, 6). A multiple is the number times a whole number, so multiples are equal to or bigger than it (the multiples of 6 are 6, 12, 18, 24...). Factors fit inside; multiples march past.

How do I know when I have found all the factors of a number?

Write each factor next to its pair: 1 with the largest, 2 with the next, and so on. The two numbers in each pair move closer together, and once they meet in the middle you have them all. For 36 the middle pair is 6x6, so you never need to test past 6.

Is 1 a factor of every number?

Yes. Every whole number can be written as 1 times itself, so 1 and the number itself are always factors. That means every number has at least two factors, except 1, which has only itself.

Can a number be both a factor and a multiple of another number?

A number is always a factor and a multiple of itself, since 6 divides 6 and 6 = 6x1. Between two different numbers, though, it only goes one way: 3 is a factor of 12, and 12 is a multiple of 3.

Do multiples ever stop?

No. You can always multiply by the next whole number, so the list of multiples goes on forever: 9, 18, 27, 36, 45, and beyond. Factors are different — every number has only a limited set of factors.