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Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 155 · 40 questions
TalentJR

How to solve it (Factors come in pairs): Find factors in pairs: 1 × N, 2 × …, 3 × …, until the pairs meet. Prime factorisation: keep dividing by the smallest prime that works, then group equal primes as powers. 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.

NameDateTime takenScore ___ / 40
  1. 1.What is the prime factorisation of 210? (a) 2 × 3 × 5 × 7² (b) 2³ × 3 × 5 × 7 (c) 2 × 3 × 5 × 7 (d) 2² × 3 × 5 × 7
  2. 2.What is the prime factorisation of 12? (a) 2³ × 3 (b) 2 × 3 (c) 2² × 3 (d) 2 × 3²
  3. 3.What is the smallest prime factor of 21? (a) 2 (b) 7 (c) 3 (d) 1
  4. 4.How many prime numbers are there between 70 and 100? (a) 8 (b) 5 (c) 7 (d) 6
  5. 5.What is the smallest prime factor of 35? (a) 2 (b) 5 (c) 1 (d) 7
  6. 6.How many prime numbers are there between 70 and 90? (a) 6 (b) 7 (c) 5 (d) 4
  7. 7.What is the smallest prime factor of 187? (a) 1 (b) 11 (c) 17 (d) 2
  8. 8.How many factors does 28 have? (a) 5 (b) 4 (c) 6 (d) 7
  9. 9.How many prime numbers are there between 30 and 50? (a) 5 (b) 7 (c) 4 (d) 6
  10. 10.How many prime numbers are there between 40 and 50? (a) 2 (b) 5 (c) 4 (d) 3
  11. 11.How many factors does 96 have? (a) 13 (b) 12 (c) 11 (d) 10
  12. 12.How many factors does 14 have? (a) 4 (b) 5 (c) 3 (d) 2
  13. 13.What is the prime factorisation of 48? (a) 2 × 3⁴ (b) 2⁵ × 3 (c) 2⁴ × 3 (d) 2³ × 3
  14. 14.How many factors does 19 have? (a) 1 (b) 3 (c) 2 (d) 4
  15. 15.How many factors does 112 have? (a) 9 (b) 10 (c) 8 (d) 11
  16. 16.What is the smallest prime factor of 209? (a) 11 (b) 19 (c) 1 (d) 2
  17. 17.How many factors does 87 have? (a) 5 (b) 2 (c) 4 (d) 3
  18. 18.What is the prime factorisation of 144? (a) 2² × 3⁴ (b) 2⁴ × 3² (c) 2³ × 3² (d) 2⁵ × 3²
  19. 19.How many factors does 49 have? (a) 1 (b) 3 (c) 4 (d) 2
  20. 20.How many factors does 33 have? (a) 3 (b) 4 (c) 5 (d) 2
  21. 21.How many factors does 95 have? (a) 2 (b) 5 (c) 4 (d) 3
  22. 22.How many prime numbers are there between 20 and 30? (a) 1 (b) 2 (c) 3 (d) 4
  23. 23.What is the prime factorisation of 28? (a) 2 × 7² (b) 2 × 7 (c) 2² × 7 (d) 2³ × 7
  24. 24.What is the smallest prime factor of 319? (a) 11 (b) 2 (c) 1 (d) 29
  25. 25.How many prime numbers are there between 20 and 40? (a) 4 (b) 5 (c) 3 (d) 6
  26. 26.What is the prime factorisation of 56? (a) 2 × 7³ (b) 2² × 7 (c) 2⁴ × 7 (d) 2³ × 7
  27. 27.What is the prime factorisation of 360? (a) 2 × 3² × 5³ (b) 2² × 3² × 5 (c) 2⁴ × 3² × 5 (d) 2³ × 3² × 5
  28. 28.How many factors does 34 have? (a) 3 (b) 5 (c) 2 (d) 4
  29. 29.What is the prime factorisation of 14? (a) 2² × 7 (b) 2³ × 7 (c) 2 × 7 (d) 2 × 7²
  30. 30.How many factors does 99 have? (a) 6 (b) 4 (c) 7 (d) 5
  31. 31.How many factors does 70 have? (a) 8 (b) 6 (c) 9 (d) 7
  32. 32.How many factors does 81 have? (a) 5 (b) 6 (c) 3 (d) 4
  33. 33.What is the prime factorisation of 126? (a) 2 × 3² × 7 (b) 2³ × 3² × 7 (c) 2² × 3² × 7 (d) 2 × 3² × 7²
  34. 34.What is the smallest prime factor of 299? (a) 13 (b) 2 (c) 1 (d) 23
  35. 35.What is the smallest prime factor of 133? (a) 7 (b) 2 (c) 1 (d) 19
  36. 36.How many factors does 89 have? (a) 4 (b) 2 (c) 1 (d) 3
  37. 37.What is the prime factorisation of 280? (a) 2² × 5 × 7 (b) 2⁴ × 5 × 7 (c) 2 × 5 × 7³ (d) 2³ × 5 × 7
  38. 38.What is the smallest prime factor of 87? (a) 1 (b) 3 (c) 2 (d) 29
  39. 39.What is the smallest prime factor of 119? (a) 1 (b) 2 (c) 17 (d) 7
  40. 40.How many factors does 88 have? (a) 6 (b) 7 (c) 9 (d) 8

Answer key

  1. 2 × 3 × 5 × 7
  2. 2² × 3
  3. 3
  4. 6
  5. 5
  6. 5
  7. 11
  8. 6
  9. 5
  10. 3
  11. 12
  12. 4
  13. 2⁴ × 3
  14. 2
  15. 10
  16. 11
  17. 4
  18. 2⁴ × 3²
  19. 3
  20. 4
  21. 4
  22. 2
  23. 2² × 7
  24. 11
  25. 4
  26. 2³ × 7
  27. 2³ × 3² × 5
  28. 4
  29. 2 × 7
  30. 6
  31. 8
  32. 5
  33. 2 × 3² × 7
  34. 13
  35. 7
  36. 2
  37. 2³ × 5 × 7
  38. 3
  39. 7
  40. 8

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