Prime factorisation, HCF and LCM

2026-10-17Numbers & Estimation Mr Chan

Write numbers as products of primes, then read off HCF and LCM — a frequent TSA-style opener.

Key formulas

HCF: \text{take lowest powers of common primes}

LCM: \text{take highest powers of all primes}

Prime factorisation

Break a whole number into a product of primes, often written with indices: 180 = 2² × 3² × 5. Listing factor trees or repeated division both work; the final form must use only primes.

HCF from prime factors

For two or more numbers, the highest common factor uses only the primes that appear in every factorisation, each raised to the lowest power that appears. Example style (numbers changed from typical papers): HCF of 2²×3 and 2×3²×5 is 2×3 = 6.

LCM from prime factors

The lowest common multiple uses every prime that appears in any factorisation, each raised to the highest power. LCM of 2²×3 and 2×3²×5 is 2²×3²×5 = 180.

Exam habit

Write each number in prime-factor form on its own line before comparing powers. Check: HCF × LCM = product of the two numbers when they are positive integers.

Worked example

HCF and LCM from primes

Problem. Find the HCF and LCM of 84 and 90.

Solution. 84 = 2²×3×7, 90 = 2×3²×5. HCF = 2×3 = 6. LCM = 2²×3²×5×7 = 1260. Check: 84×90 = 7560 and 6×1260 = 7560.

Review quiz

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HCF of 2²×3 and 2×3²×5 is

Practice questions

Express 252 as a product of prime factors.
Find the HCF and LCM of 84 and 90.
Find the HCF of 2³×5² and 2×5×7.
Two numbers are 2²×3×11 and 2×3³. Find their LCM.

Study notes

Is 1 a prime?

No. Prime factorisations never include 1 as a factor.

What if a prime is missing from one number?

It contributes power 0 to the HCF (so it is omitted) but its full power to the LCM.