Factors, multiples and prime numbers
National Curriculum: this is in the Year 5 programme of study, part of key stage 2 (Years 3 to 6). Schools have to teach the key stage 2 programme by the end of Year 6, and can introduce content earlier or later within it.
Children in Year 5 are 9 or 10, and turn 10 during the school year.
What it is
A factor of a number divides into it exactly, with nothing left over: the factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple is a number you land on counting up in steps of that number: 12, 24, 36 and 48 are multiples of 12. The two words describe one fact from opposite ends, so 3 × 4 = 12 says that 3 and 4 are factors of 12 and that 12 is a multiple of both. A factor is never bigger than its number and a multiple is never smaller, which is the quickest way to tell which word a question is using.
A prime number has exactly two factors: itself and 1. So 7 is prime, 1 is not prime because it has only one factor, and 2 is prime and is the only even prime, because every other even number has 2 as a factor.
Testing a number below 100 takes four divisions: by 2, by 3, by 5 and by 7. If none of them goes in exactly, the number is prime, because a number below 100 that is not prime always has one of those four as a factor. That test is what separates the two numbers that catch most people out: 51 is 3 × 17 and 91 is 7 × 13, so neither is prime.
When children learn it
Year 5 is where the National Curriculum asks children to identify multiples and factors, find all the factor pairs of a number and the common factors of two numbers, use the words prime number, prime factor and composite, establish whether a number up to 100 is prime, and recall the prime numbers up to 19. Year 4 leads in with factor pairs. Year 6 asks for common factors, common multiples and primes together, and factors do much of the work when fractions are simplified.
What it looks like when it goes wrong
These patterns come up again and again in children's answers on this skill. Each one points at a specific missing idea, not carelessness.
- A multiple offered where a factor was asked. Asked for a factor of 20, they answer 40. The two words are being held as one idea rather than as opposite directions. Asking which of the two can be bigger than 20 sorts it out faster than a definition does.
- An odd number taken to be prime. Asked which of a set of numbers is prime, they choose 51 or 91. Being odd is being used as the test. Both of those are odd and neither is prime, and dividing by 3 and by 7 is what tells them apart from a real prime such as 53.
- A near number offered as a factor. Asked for a factor of 30, they answer 7 or 8. The answer is being judged by size rather than tested: 30 ÷ 7 leaves 2 over and 30 ÷ 8 leaves 6. A factor is a division that comes out exactly, so the test is to do the division.
How to help at home
- Test rather than guess. Does it divide exactly, with nothing left over? A factor is the answer to a division question, not a feeling about the right sort of size.
- Learn three quick checks: every even number divides by 2, a number whose digits add up to a multiple of 3 divides by 3, and a number ending in 0 or 5 divides by 5.
- Ask for the primes up to 19 now and then: 2, 3, 5, 7, 11, 13, 17 and 19. Year 5 asks for those from memory, and having them means most prime questions are already answered.
- Take one table fact and ask it both ways: 6 × 7 = 42, so which numbers are the factors and which is the multiple? Saying both keeps the two words apart.