FactorsWhat are factors and multiples?
A factor of a number is a whole number that divides it exactly, leaving no remainder. A multiple of a number is what you get when you multiply it by a whole number. The two words describe the same fact from opposite ends: since 3 × 4 = 12, the numbers 3 and 4 are factors of 12, and 12 is a multiple of both 3 and 4.
Finding the factors of a number
Take 36. Test the whole numbers in order and write each one that divides with the number it pairs with.
- 1 × 36
- 2 × 18
- 3 × 12
- 4 × 9
- 5 does not divide 36
- 6 × 6
Stop at 6, because 6 × 6 is 36 itself. Any larger factor has already appeared as the partner of a smaller one. So the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. In general you need to test only up to the square root of the number.
Facts about factors
- 1 is a factor of every number, and every number is a factor of itself.
- Every factor of a number is less than or equal to the number.
- A number has a limited count of factors. 36 has nine.
- Factors come in pairs. Only a perfect square has a pair made of one number twice, so only perfect squares have an odd count of factors.
Finding multiples
The multiples of 7 are 7, 14, 21, 28, 35 and so on: the table of 7, going on for ever.
- Every number is a multiple of itself and of 1.
- Every multiple of a number is greater than or equal to the number.
- A number has endless multiples. There is no largest one.
How to tell them apart
Children mix the two words up more than any others in this chapter. A short memory aid: factors are few and small, multiples are many and large. Factors fit inside the number. Multiples are built out of it.
| Factors of 12 | Multiples of 12 | |
|---|---|---|
| What they are | 1, 2, 3, 4, 6, 12 | 12, 24, 36, 48, 60 and so on |
| How many | Six | Endless |
| Size | 12 or less | 12 or more |
| How to find | Divide 12 by 1, 2, 3 and so on | Multiply 12 by 1, 2, 3 and so on |
Common factors and common multiples
A number that is a factor of two numbers is a common factor. The factors of 12 are 1, 2, 3, 4, 6, 12 and those of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3 and 6, and the largest, 6, is the highest common factor.
A number that is a multiple of two numbers is a common multiple. Multiples of 4 are 4, 8, 12, 16, 20, 24 and of 6 are 6, 12, 18, 24. The common multiples are 12, 24, 36 and so on, and the smallest, 12, is the lowest common multiple.
Where factors and multiples are used
- Sharing equally. 24 sweets can be shared equally among 1, 2, 3, 4, 6, 8, 12 or 24 children: the factors of 24.
- Arranging in rows. 30 chairs can be set in rows of 5 or 6 but not in rows of 4.
- Fractions. Simplifying 18/24 uses the common factor 6 to get 3/4. Adding 1/4 and 1/6 uses the common multiple 12.
- Things that repeat. A bus every 12 minutes and another every 18 minutes leave together at common multiples of 12 and 18.
Use the factor finder for any number, or read on about prime numbers, the building blocks of factors.