Prime factors, HCF and LCM
Maths · Key Stage 3 · The School
Every number breaks into primes, once only
Split 60 repeatedly until only primes remain: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5. Written with powers that is 2² × 3 × 5. Start the tree differently — 60 = 6 × 10 — and you land on exactly the same set. That uniqueness is the reason prime factors are worth the effort: the answer does not depend on how you began.
The HCF is what both numbers share
For 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7, take the primes appearing in both, each at the LOWER power: 2² × 3 = 12. Twelve is the largest number dividing both. Listing every factor of both and comparing gets the same answer and takes ten times as long once the numbers grow.
The LCM is everything either one needs
Same two numbers, but now take every prime that appears in either, each at the HIGHER power: 2² × 3 × 5 × 7 = 420. That is the first number both 60 and 84 divide into. Lower power for HCF, higher power for LCM — one word apart, and the whole difference between the two answers.
Where this is actually used
Two buses leave at 8am, one every 60 minutes and one every 84. When do they next leave together? The LCM: 420 minutes, so 3pm. Cutting a 60cm and an 84cm ribbon into equal whole-centimetre pieces with none wasted? The HCF: 12cm. If the question says “at the same time again” it wants LCM; if it says “largest equal pieces” it wants HCF.
Why the prime factors give both answers
Once two numbers are broken into primes, the HCF and LCM fall out of the same working. For 24 = 2×2×2×3 and 36 = 2×2×3×3: the factors they SHARE are 2×2×3 = 12, which is the highest common factor. Take everything either number needs — 2×2×2×3×3 = 72 — and that is the lowest common multiple. A useful check: HCF × LCM always equals the product of the two numbers, and 12 × 72 = 864 = 24 × 36. If that does not hold, one of your factor trees is wrong, and it is worth doing before writing the answer down.