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Factors and Multiples

In one line: A factor is a whole number that divides exactly into another number with nothing left over, and a multiple is the answer you get when you multiply a number by a whole number, like the answers in a times table.

Why it works

Factors and multiples help you share things out equally, simplify fractions, find common denominators (the same bottom number) when adding fractions, and spot patterns in numbers. They also make mental maths faster, because knowing that 4 × 6 = 24 tells you about factors of 24 and multiples of 4 and 6 all at once.

Think of it like: Imagine you have 12 chairs and want to set them out in equal rows with no chairs left over. You could make 1 row of 12, 2 rows of 6, 3 rows of 4, 4 rows of 3, 6 rows of 2 or 12 rows of 1. The row sizes that work (1, 2, 3, 4, 6, 12) are the factors of 12. Now flip it around: if every row has 3 chairs, then 1 row, 2 rows, 3 rows and so on give you 3, 6, 9, 12 chairs. Those totals are the multiples of 3. (Where this breaks: The chairs picture only works for whole numbers of chairs, so it cannot show fractions or negative numbers. It also hides the fact that multiples go on forever: you could never set out every multiple of 3 in chairs, but the list 3, 6, 9, 12… never ends.)

Worked example

Find all the factors of 24 and the first five multiples of 6. Is 6 a factor of 24, and is 24 a multiple of 6?

  1. Start with 1: 1 × 24 = 24, so 1 and 24 are a factor pair (two numbers that multiply to make 24).
  2. Try 2, 3 and 4 in order: 2 × 12 = 24, 3 × 8 = 24 and 4 × 6 = 24, so these are all factor pairs.
  3. Try 5: 24 ÷ 5 = 4 remainder 4, so 5 is not a factor. The next number, 6, is already in a pair, so you can stop.
  4. List the factors in order: 1, 2, 3, 4, 6, 8, 12, 24.
  5. Multiply 6 by 1, 2, 3, 4 and 5 to get the first five multiples of 6: 6, 12, 18, 24, 30.
  6. Both lists include the link 6 × 4 = 24, so 6 is a factor of 24 and 24 is a multiple of 6.

Answer: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The first five multiples of 6 are 6, 12, 18, 24 and 30. Yes: 6 is a factor of 24, and 24 is a multiple of 6.

Common mistakes

  • Mixing up the two words is common, so remember that factors fit into a number (they are never bigger than it) while multiples come from multiplying up (they are never smaller than it, for counting numbers).
  • Forgetting 1 and the number itself is easy to do, so always start your factor list with the pair 1 × the number.
  • Missing factors by guessing at random happens a lot, so test 1, 2, 3, 4… in order and stop once the pairs start repeating.
  • Thinking a number is not a multiple of itself is wrong, because 7 × 1 = 7, so 7 is the first multiple of 7.

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Which of these numbers is a factor of 18?

Questions parents ask

What year do children learn factors and multiples?

In England, children meet factor pairs and multiples in Year 4 and study them formally in Years 5 and 6, including common factors and common multiples. In Australia this is mainly covered in Years 5 and 6, and in the US it is introduced in Grade 4, with greatest common factor and least common multiple coming in Grade 6.

How can I help my child with factors and multiples at home?

Use small objects like buttons or coins and ask your child to arrange a number, such as 12 or 20, into equal rows to find its factors. For multiples, practise counting in steps (3, 6, 9, 12…) during car journeys or while setting the table. Strong times-table knowledge makes both ideas much easier.

What is the easiest way to remember the difference between factors and multiples?

Try the phrase 'factors fit, multiples multiply'. Factors fit exactly into a number, so there are only a few of them, while multiples are made by multiplying, so they keep going forever.

Is 1 a factor of every number?

Yes, 1 is a factor of every whole number because any number divided by 1 leaves no remainder. Every whole number bigger than 1 also has itself as a factor, which is why these two factors should always be at the start and end of a factor list.