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Mathematics · Factors and multiples

Prime and composite numbers

Why are some numbers impossible to split into equal groups?

Factors and multiples, prime and composite numbers, the Sieve of Eratosthenes, divisibility tests, twin primes and co-primes.

5 depths≈ 245 min of reading79 practice questions

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  1. Discoverabout 35 minNumbers that will not make rectanglesShare laddoos, set out chairs and build rectangles from tiles to meet factors and multiples. Discover prime numbers, composite numbers, the odd case of 1, the Sieve of Eratosthenes, twin primes and co-primes.Owner-accepted · not line-by-line reviewed
  2. Understandabout 45 minFactors, primes and how to test themFind every factor with the factor-pair method, sieve to 100 and see why you can stop at 7, test any number for primality by trial division up to its square root, use divisibility rules, and meet twin primes, co-primes and factor trees.Owner-accepted · not line-by-line reviewed
  3. Investigateabout 50 minHunting patterns among the primesTest claims about primes like a mathematician: how fast primes thin out, the 6-column grid, last digits, twin prime hunts, why 3, 5, 7 stands alone, co-prime experiments, patterns that break, prime deserts and numbers with the most factors.Owner-accepted · not line-by-line reviewed
  4. Go deeperabout 55 minWhy it all works: proofs about primesProve that every number is built from primes in exactly one way, see a world where that fails, count factors from a factorisation, explain the square-root rule and every divisibility test, follow Euclid’s proof that primes never end, and prove facts about co-primes and twin primes.Owner-accepted · not line-by-line reviewed
  5. Extendabout 60 minPrimes in the wild: cicadas, codes and unsolved puzzlesTake primes into the world: prime cicada cycles, the prime-based codes behind online payments, Mersenne primes and perfect numbers, Goldbach’s and the twin prime conjectures, Indian mathematicians, other number bases, olympiad puzzles and projects.Owner-accepted · not line-by-line reviewed

Practise and apply

Question bank · 79 questions

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

Foundation 20Core 28Stretch 22Challenge 9

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Connected ideas

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

Prime and composite numbers

  • Builds on

    Four operations

    Testing whether a number is prime is just careful division: does anything divide it exactly?

  • Helps you understand

    HCF and LCM

    Prime factorisation is the fastest route to both the HCF and the LCM.

  • Contrasts with

    Number and shape patterns

    Primes famously refuse to follow a simple pattern, unlike even numbers, squares or multiples.

  • Used inanother area

    Electricity

    The encryption that protects smart meters and grid control systems relies on the difficulty of factorising huge numbers into primes.

Words to know

67 words are explained in these lessons

All vocabulary →

factormultipledivides exactlyfactor pairremainderprime numbercomposite numberneither prime nor compositetwin primesco-prime numbersSieve of Eratostheneseven number+55 more