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Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 105 · 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 35? (a) 5² × 7 (b) 5 × 7² (c) 5³ × 7 (d) 5 × 7
  2. 2.How many factors does 59 have? (a) 4 (b) 2 (c) 3 (d) 1
  3. 3.How many factors does 57 have? (a) 3 (b) 5 (c) 4 (d) 2
  4. 4.What is the prime factorisation of 14? (a) 2 × 7² (b) 2³ × 7 (c) 2 × 7 (d) 2² × 7
  5. 5.What is the prime factorisation of 252? (a) 2 × 3² × 7 (b) 2² × 3² × 7 (c) 2³ × 3² × 7 (d) 2 × 3² × 7²
  6. 6.How many prime numbers are there between 10 and 20? (a) 5 (b) 3 (c) 6 (d) 4
  7. 7.What is the prime factorisation of 105? (a) 3 × 5 × 7 (b) 3 × 5 × 7² (c) 3³ × 5 × 7 (d) 3² × 5 × 7
  8. 8.How many factors does 89 have? (a) 2 (b) 1 (c) 3 (d) 4
  9. 9.What is the smallest prime factor of 217? (a) 31 (b) 2 (c) 1 (d) 7
  10. 10.What is the prime factorisation of 280? (a) 2⁴ × 5 × 7 (b) 2³ × 5 × 7 (c) 2² × 5 × 7 (d) 2 × 5 × 7³
  11. 11.How many prime numbers are there between 50 and 60? (a) 3 (b) 1 (c) 2 (d) 4
  12. 12.What is the smallest prime factor of 209? (a) 2 (b) 19 (c) 1 (d) 11
  13. 13.How many factors does 93 have? (a) 3 (b) 2 (c) 5 (d) 4
  14. 14.What is the prime factorisation of 360? (a) 2 × 3² × 5³ (b) 2⁴ × 3² × 5 (c) 2² × 3² × 5 (d) 2³ × 3² × 5
  15. 15.How many prime numbers are there between 40 and 50? (a) 2 (b) 3 (c) 4 (d) 5
  16. 16.How many factors does 32 have? (a) 7 (b) 4 (c) 6 (d) 5
  17. 17.How many factors does 58 have? (a) 5 (b) 2 (c) 3 (d) 4
  18. 18.How many factors does 85 have? (a) 3 (b) 2 (c) 5 (d) 4
  19. 19.How many prime numbers are there between 80 and 90? (a) 4 (b) 1 (c) 2 (d) 3
  20. 20.How many factors does 86 have? (a) 2 (b) 5 (c) 4 (d) 3
  21. 21.How many factors does 110 have? (a) 7 (b) 8 (c) 6 (d) 9
  22. 22.How many prime numbers are there between 30 and 40? (a) 2 (b) 1 (c) 4 (d) 3
  23. 23.What is the prime factorisation of 140? (a) 2 × 5 × 7² (b) 2³ × 5 × 7 (c) 2 × 5 × 7 (d) 2² × 5 × 7
  24. 24.What is the prime factorisation of 40? (a) 2³ × 5 (b) 2² × 5 (c) 2⁴ × 5 (d) 2 × 5³
  25. 25.What is the smallest prime factor of 119? (a) 7 (b) 1 (c) 2 (d) 17
  26. 26.How many prime numbers are there between 70 and 90? (a) 4 (b) 6 (c) 7 (d) 5
  27. 27.What is the smallest prime factor of 121? (a) 11 (b) 2 (c) 1 (d) 3
  28. 28.How many factors does 109 have? (a) 1 (b) 2 (c) 4 (d) 3
  29. 29.How many factors does 35 have? (a) 3 (b) 2 (c) 5 (d) 4
  30. 30.What is the prime factorisation of 28? (a) 2 × 7 (b) 2 × 7² (c) 2² × 7 (d) 2³ × 7
  31. 31.How many factors does 38 have? (a) 2 (b) 5 (c) 3 (d) 4
  32. 32.How many prime numbers are there between 30 and 60? (a) 9 (b) 7 (c) 6 (d) 8
  33. 33.How many factors does 37 have? (a) 2 (b) 1 (c) 4 (d) 3
  34. 34.How many factors does 81 have? (a) 6 (b) 4 (c) 5 (d) 3
  35. 35.How many prime numbers are there between 90 and 110? (a) 6 (b) 4 (c) 7 (d) 5
  36. 36.How many factors does 105 have? (a) 8 (b) 6 (c) 7 (d) 9
  37. 37.What is the prime factorisation of 56? (a) 2 × 7³ (b) 2⁴ × 7 (c) 2² × 7 (d) 2³ × 7
  38. 38.How many factors does 69 have? (a) 4 (b) 2 (c) 5 (d) 3
  39. 39.What is the smallest prime factor of 161? (a) 2 (b) 1 (c) 7 (d) 23
  40. 40.What is the smallest prime factor of 39? (a) 13 (b) 2 (c) 1 (d) 3

Answer key

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

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