Factors, Multiples and Primes
Factors go into a number. Multiples come out of it. Children mix them up constantly, and the fix is a picture rather than a definition.
Factors and multiples
A factor divides into a number exactly. A multiple is what you get when you multiply it.
The factors of 12 are 1, 2, 3, 4, 6 and 12, because each of those divides 12 with nothing left over. The multiples of 12 are 12, 24, 36, 48 and onwards forever. So factors are a short list that stops, and multiples are an endless one that starts at the number itself.
Which way round
Factors are smaller than or equal to the number, and there are only a few. Multiples are bigger than or equal to it, and they never run out.
If a child gives an answer bigger than the number when asked for factors, that alone tells them they have swapped the two.
Finding every factor
Factors come in pairs, because every rectangle has two sides. 24 counters can be laid out as 4 rows of 6, which means 4 and 6 are both factors of 24 and they arrive together.
That gives a method that does not miss any. Start at 1 and work upwards, and each time you find a factor, write down its partner as well. Stop when the two numbers in the pair meet.
Find all the factors of 24.
- 1 x 24. Write both.
- 2 x 12. Write both.
- 3 x 8. Write both. Does 4 go? Yes: 4 x 6. Write both.
- Next would be 5, which does not go, then 6, but 6 is already on the list as the partner of 4. The pairs have met, so stop.
1, 2, 3, 4, 6, 8, 12, 24
Prime numbers
A prime number has exactly two factors: 1 and itself.
7 is prime because nothing divides it except 1 and 7. Put another way, seven counters can only be laid out as one row of seven, and no other rectangle is possible. That is what being prime looks like: a number with no interesting rectangles.
The primes up to 20 are 2, 3, 5, 7, 11, 13, 17 and 19. It is worth knowing them by sight, because they turn up constantly once fractions start needing simplifying.
2 is the odd one out, by being even
2 is the only even prime, because every other even number has 2 as a factor and so has at least three factors. Children often assume prime means odd, which then makes 9, 15 and 21 look prime when none of them is.
Why 1 is not prime
This looks like a technicality invented to be annoying, and it is worth explaining rather than asserting.
1 has only one factor, itself, and the definition asks for exactly two. But the deeper reason is that every whole number breaks down into primes in exactly one way: 12 is 2 x 2 x 3 and there is no other prime recipe for it. If 1 counted as prime, you could write 1 x 2 x 2 x 3, or 1 x 1 x 2 x 2 x 3, and the word "exactly" would stop being true. Leaving 1 out keeps the rest of the subject tidy.
Common mistakes
1. Swapping factors and multiples
Being asked for factors of 6 and answering 12, 18, 24. The check is size: a factor is never bigger than the number. Say it before answering and the mix-up disappears, because the two lists point in opposite directions.
2. Missing factors by not working in pairs
Listing the factors of 36 as 1, 2, 3, 4, 6, 36 and stopping. Working in pairs would have caught 9, 12 and 18 automatically, because each is the partner of one already on the list. Hunting one at a time is how factors get missed.
3. Believing every odd number is prime
Calling 9, 15, 21 or 27 prime. All of them are odd and none is prime, because 9 is 3 x 3, 15 is 3 x 5, 21 is 3 x 7 and 27 is 3 x 9. Checking whether 3 divides it catches almost every one of these.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
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What to learn next
Factors are what simplifying a fraction actually uses: you are dividing top and bottom by a common factor, and finding the biggest one makes it a single step.
For parents
Give your child a number of counters, or coins, or pasta shapes, and ask them to make every rectangle they can. Twelve gives three rectangles, thirteen gives one. They will discover prime numbers themselves in about four minutes, and a discovery is remembered in a way a definition is not.
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