Highest Common Factor and Lowest Common Multiple made simple by reading them straight off the prime factors — with a Venn-diagram picture, exam-style worked examples and word problems, plus a worksheet with memo.
HCF and LCM scare a lot of learners, but once you can prime factorise (the previous lesson) they become almost automatic. They are just two different ways of reading the same set of primes.
1Highest Common Factor (HCF)
From the prime factors, take only the primes the numbers share.
2Lowest Common Multiple (LCM)
From the prime factors, take every prime that appears, each at its highest power.
3Reading both off the prime factors
- 112 = 2 × 2 × 3 and 18 = 2 × 3 × 3.Prime factorise both first — every time.
- 2Shared primes: one 2 and one 3 → HCF = 2 × 3 = 6.HCF uses only what they have in common.
- 3All primes at highest power: 22 × 32 = 4 × 9 → LCM = 36.The LCM must contain every prime so both divide into it.
Classic slip: giving the HCF of 12 and 18 as 2 (stopping at the first shared factor). Use all the shared primes — 2 and 3 — so the HCF is 6, not 2.
4Exam word problems — which one do I use?
Splitting or sharing into equal groups → HCF. Events repeating and lining up together → LCM. Deciding correctly is where most of the marks are.
- 1One light flashes every 12 s, the other every 18 s. They flash together now — when next?"Together again" = a common multiple → LCM.
- 2LCM of 12 and 18 = 36.Smallest time both complete whole cycles.
- 3They flash together again after 36 seconds.Always answer in words.
- 1Two ribbons, 24 cm and 60 cm, are cut into equal pieces as long as possible, none left over. How long is each piece?"Equal pieces, largest possible" = HCF.
- 224 = 23 × 3, 60 = 22 × 3 × 5 → HCF = 22 × 3 = 12.Take only the shared primes.
- 3Each piece is 12 cm long.Answer in words.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Find the HCF of 24 and 36.
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24 = 23×3, 36 = 22×32 → shared 22×3 = 12 - 2Find the LCM of 8 and 12.
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8 = 23, 12 = 22×3 → 23×3 = 24 - 3Find the HCF and LCM of 18 and 30.
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18 = 2×32, 30 = 2×3×5 → HCF = 2×3 = 6; LCM = 2×32×5 = 90 - 4Two bells ring every 15 min and every 20 min. They ring together at 08:00. When do they next ring together?
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LCM of 15 & 20 = 60 min → 09:00 - 560 red and 48 blue beads are shared into identical packs (same reds per pack, same blues per pack) using as many packs as possible. How many packs?
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Number of packs = HCF(60, 48). 60 = 22×3×5, 48 = 24×3 → HCF = 22×3 = 12 packs
Quick Quiz
4 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 8 past papers and worksheets on this topic.