Factors, multiples and primes
Maths · Key Stage 2 · The School
Factors divide it, multiples come from it
A factor of 12 divides into 12 leaving nothing over: 1, 2, 3, 4, 6, 12. A multiple of 12 is what you get by multiplying 12 by something: 12, 24, 36 and on forever. Factors are a short list that stops; multiples are an endless one. Mixing them up is the single most common error in this topic.
Hunt factors in pairs so none escape
Start at 1 and walk upwards, writing each pair as you find it: 1 × 24, 2 × 12, 3 × 8, 4 × 6. Then try 5 — it does not divide 24 — and then you reach 6, which you already have. The moment the pair turns back on itself, you are finished, and you can be certain nothing was missed.
A prime has exactly two factors
Seven has 1 and 7 and nothing else, so it is prime. Nine has 1, 3 and 9 — three factors — so it is not. The primes below twenty are 2, 3, 5, 7, 11, 13, 17 and 19. Two is the only even prime, which feels wrong and is simply true: every other even number has 2 as a spare factor.
Why one is not prime
One has exactly one factor, itself, and the definition demands two. This is not a technicality invented to be annoying. Every whole number breaks into primes in exactly one way, and if 1 counted as prime you could pad any answer with as many 1s as you liked and that uniqueness would collapse.
Why 1 is not prime, and why that is a decision
A prime has exactly two factors: itself and 1. The number 1 has only one factor — itself — so it does not qualify, and that is not a technicality anyone invented to be awkward. If 1 counted as prime, then 12 could be written as 2×2×3, or 1×2×2×3, or 1×1×2×2×3, and the whole point of prime factorisation is that every number breaks down in exactly one way. Excluding 1 is what keeps that true. It is a good example of a definition chosen to make later mathematics work rather than handed down from nowhere.