Factors, multiples, primes — and finding HCF and LCM

This topic underpins a surprising amount of GCSE maths. You'll meet it again in surds, fractions, ratios, and even algebraic factorisation. Get fluent with the prime-factorisation method and the rest becomes mechanical.

Core vocabulary

Factor — a whole number that divides exactly into another with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12.

Multiple — what you get when you multiply a number by an integer. The first five multiples of 7 are 7, 14, 21, 28 and 35.

Prime number — a number with exactly two factors: 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note 1 is not prime; 2 is the only even prime.

Composite number — anything that has more than two factors (4, 6, 8, 9, 10, 12, …).

Prime factorisation — the big idea

Every integer above 1 can be written as a unique product of primes. This is the Fundamental Theorem of Arithmetic and it's the engine behind both HCF and LCM.

Two reliable methods:

Method 1: Factor trees Keep splitting until every leaf is prime. Example with 60:

        60
       /  \
      6    10
     / \  / \
    2  3 2  5

So 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Method 2: Repeated division Divide by the smallest prime that fits, write what's left, repeat: 60 ÷ 2 = 30; 30 ÷ 2 = 15; 15 ÷ 3 = 5; 5 ÷ 5 = 1. So 60 = 2² × 3 × 5.

HCF (Highest Common Factor)

After prime-factorising both numbers, the HCF is the product of the primes appearing in BOTH lists, taken to the LOWER power.

Example: HCF of 60 and 84.

• 60 = 2² × 3 × 5
• 84 = 2² × 3 × 7

Shared primes: 2 (lower power: 2²) and 3 (lower power: 3¹). HCF = 2² × 3 = 12.

LCM (Lowest Common Multiple)

After prime-factorising, the LCM is the product of EVERY prime that appears in EITHER list, taken to the HIGHER power.

Continuing with 60 and 84: Primes appearing: 2 (higher power 2²), 3 (higher power 3¹), 5 (higher power 5¹), 7 (higher power 7¹). LCM = 2² × 3 × 5 × 7 = 420.

Useful identity

For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b.

This is a fast sanity check. With 60 and 84: 12 × 420 = 5040; 60 × 84 = 5040. ✓

When this comes up

• Cancelling fractions to lowest terms: divide top and bottom by their HCF.
• Adding fractions: use the LCM of the denominators.
• Worded LCM problems ("two buses leave together — when do they next coincide?"): the answer is the LCM of the intervals.
• Worded HCF problems ("largest tile that fits both rooms exactly"): HCF of the two dimensions.