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Mathematics · Factors and multiples

HCF and LCM

When will two blinking lights flash together again — and what is the biggest tile that fits a floor exactly?

Highest common factor and lowest common multiple by listing, prime factorisation and division, their link HCF × LCM = product, and real problems that need them.

5 depths≈ 240 min of reading79 practice questions

The ladder

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  1. Discoverabout 35 minSharing and meeting: meet the HCF and LCMStart from two puzzles, the biggest tile for a courtyard and the next time two lights flash together, and discover factors, multiples, common factors, common multiples, the HCF and the LCM, and how to tell which one a problem needs.Owner-accepted · not line-by-line reviewed
  2. Understandabout 45 minFour ways to find the HCF and LCMPrecise definitions, then four methods: listing, prime factorisation with a Venn picture, long (continued) division for the HCF and common division for the LCM. Three numbers, the rule HCF × LCM = product, co-primes, fractions and the classic mix-ups.Owner-accepted · not line-by-line reviewed
  3. Investigateabout 45 minPredict, test and explain: HCF and LCM patternsMake predictions and test them: when the LCM equals the product, why neighbours are co-prime, how HCF × LCM = a × b holds for two numbers but not three, what scaling does, how remainder puzzles work, and how changing a word problem changes the answer.Owner-accepted · not line-by-line reviewed
  4. Go deeperabout 55 minWhy it works: proofs, Euclid and the edgesProofs in plain language: unique prime factorisation, why HCF takes smallest powers and LCM largest, why HCF × LCM = a × b (and why not for three numbers), why Euclid’s method works and how fast it is, Bézout’s identity, edge cases, harder problems and history.Owner-accepted · not line-by-line reviewed
  5. Extendabout 60 minCycles, gears and puzzles: HCF and LCM in the wildExpeditions beyond the textbook: cycles with head starts, calendars and planetary alignments (and why they are not LCMs), prime-cycle cicadas, gears and bicycle chains, tala rhythms, water jugs, ancient remainder puzzles, screen ratios, fractions, olympiad problems, careers and open questions.Owner-accepted · not line-by-line reviewed

Practise and apply

Question bank · 79 questions

8 sections from Foundation to Challenge, each with hints and a worked solution — or play a 10-question round for points and stars.

Foundation 20Core 31Stretch 18Challenge 10

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The web

Connected ideas

Follow a connection when you’re curious — some lead into completely different areas.

HCF and LCM

  • Used in

    Number and shape patterns

    Two repeating cycles line up again after their LCM — the pattern behind blinking lights and bus timetables.

  • Used in

    Shape and space

    The largest square tile that fits a rectangular floor exactly has a side equal to the HCF of its length and width.

Words to know

46 words are explained in these lessons

All vocabulary →

factordivisordivisiblemultiplecommon factorcommon multipleHCFGCD / GCFLCMremainderprime numbercomposite number+34 more