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Maths Olympiad · topic 31 of 33 · free

Primes and divisors — Maths Olympiad practice

A prime has exactly two factors: 1 and itself. Find factors in pairs so none are missed, break a number into primes by dividing by the smallest prime that works, and remember that 1 is not a prime and 2 is the only even prime.

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How to solve it

Factors come in pairs — “every factor below the square root has a partner above it”

  1. Find factors in pairs: 1 × N, 2 × …, 3 × …, until the pairs meet.
  2. Prime factorisation: keep dividing by the smallest prime that works, then group equal primes as powers.
  3. 1 is not a prime; 2 is the only even prime. To test a number, try dividing by the primes up to its square root.

Use it for: Factor, prime, HCF-style and “how many primes” questions in Class 5–9 Olympiad papers.

Worked example: How many factors does 36 have? → 9

  1. Factor pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.
  2. The factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36 — 6 is counted once, so there are 9.

Check: 36 = 2² × 3², and (2 + 1) × (2 + 1) = 9 factors ✓

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✓ Written and checked by the TalentJR team · Updated 13 Oct 2026 · How we make our lessons

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🏁 Topic 31 test

20 questions. Check each answer, then go to the next one. Get 16 or more right to pass.

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🖨️ Prime Numbers and Factors Worksheet for Olympiad

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