Why does a chocolate bar come in 12 or 24 chunks and never 13? Because 12 and 24 split evenly lots of ways, and 13 barely splits at all. This topic is about how whole numbers fit together — which numbers divide into them, and which numbers they divide into. It powers everything from sharing sweets fairly to simplifying fractions.
Why this matters: factors and multiples are the machinery behind cancelling fractions, finding common denominators, and spotting patterns. HCF and LCM turn up in real problems too — like working out when two buses that come every 6 and 8 minutes will next arrive together.
Factor tree: 60 = 2 × 2 × 3 × 5
Factors
Each rectangle of 12 squares is a factor pair — so the factors are 1, 2, 3, 4, 6, 12.
A factor of a number divides into it exactly, with nothing left over. Factors come in pairs that multiply to give the number, so hunting in pairs means you never miss one.
Factors of 12: 1×12, 2×6, 3×4 → so the factors are 1, 2, 3, 4, 6, 12. Working through 1, 2, 3… in order and writing both partners keeps the list complete.
Multiples
Multiples of 5 land every 5 steps and never stop — unlike the finite list of factors.
A multiple is what you get in the times table — multiply the number by 1, 2, 3, and so on. Unlike factors, multiples never run out.
Multiples of 5: 5, 10, 15, 20, 25, … (they never stop).
Factor vs multiple: think of a family tree. Factors are the smaller ancestors that go into a number; multiples are the bigger descendants it goes into. So 3 is a factor of 12, and 12 is a multiple of 3 — the same relationship read two ways.
Factors of 12
Multiples of 12
What
divide into 12
12 divides into them
Size
12 or smaller
12 or bigger
List
1, 2, 3, 4, 6, 12
12, 24, 36, 48, …
Prime numbers
A prime number has exactly two factors: 1 and itself. The first few are 2, 3, 5, 7, 11, 13… Primes are the "atoms" of number — every other whole number is built from them.
Watch out: 1 is not prime (it has only one factor). And 2 is the only even prime — every other even number is divisible by 2, so it has 2 as an extra factor.
Writing a number as a product of primes
Every whole number breaks down into primes multiplied together, in one unique way. Use a factor tree: keep splitting each branch until it ends on a prime, then collect them up.