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Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 40 · 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 smallest prime factor of 35? (a) 2 (b) 1 (c) 5 (d) 7
  2. 2.How many factors does 68 have? (a) 6 (b) 7 (c) 5 (d) 4
  3. 3.How many factors does 85 have? (a) 5 (b) 3 (c) 2 (d) 4
  4. 4.How many factors does 55 have? (a) 3 (b) 2 (c) 5 (d) 4
  5. 5.How many prime numbers are there between 70 and 80? (a) 4 (b) 3 (c) 2 (d) 5
  6. 6.What is the prime factorisation of 70? (a) 2² × 5 × 7 (b) 2³ × 5 × 7 (c) 2 × 5 × 7 (d) 2 × 5 × 7²
  7. 7.How many factors does 113 have? (a) 3 (b) 4 (c) 2 (d) 1
  8. 8.How many factors does 77 have? (a) 4 (b) 5 (c) 3 (d) 2
  9. 9.What is the prime factorisation of 20? (a) 2 × 5 (b) 2³ × 5 (c) 2² × 5 (d) 2 × 5²
  10. 10.What is the prime factorisation of 315? (a) 3 × 5 × 7 (b) 3³ × 5 × 7 (c) 3² × 5 × 7 (d) 3 × 5 × 7²
  11. 11.How many factors does 102 have? (a) 6 (b) 8 (c) 7 (d) 9
  12. 12.What is the prime factorisation of 15? (a) 3² × 5 (b) 3 × 5² (c) 3³ × 5 (d) 3 × 5
  13. 13.How many prime numbers are there between 30 and 50? (a) 6 (b) 4 (c) 5 (d) 7
  14. 14.What is the prime factorisation of 21? (a) 3 × 7² (b) 3 × 7 (c) 3² × 7 (d) 3³ × 7
  15. 15.How many prime numbers are there between 90 and 100? (a) 2 (b) 1 (c) 0 (d) 3
  16. 16.What is the smallest prime factor of 247? (a) 19 (b) 13 (c) 1 (d) 2
  17. 17.How many factors does 34 have? (a) 3 (b) 2 (c) 4 (d) 5
  18. 18.How many factors does 22 have? (a) 2 (b) 3 (c) 4 (d) 5
  19. 19.How many factors does 72 have? (a) 10 (b) 11 (c) 13 (d) 12
  20. 20.How many prime numbers are there between 50 and 60? (a) 2 (b) 4 (c) 3 (d) 1
  21. 21.What is the smallest prime factor of 93? (a) 3 (b) 2 (c) 1 (d) 31
  22. 22.What is the prime factorisation of 80? (a) 2³ × 5 (b) 2 × 5⁴ (c) 2⁵ × 5 (d) 2⁴ × 5
  23. 23.How many factors does 71 have? (a) 1 (b) 2 (c) 3 (d) 4
  24. 24.What is the prime factorisation of 112? (a) 2 × 7⁴ (b) 2³ × 7 (c) 2⁴ × 7 (d) 2⁵ × 7
  25. 25.How many prime numbers are there between 20 and 40? (a) 3 (b) 4 (c) 5 (d) 6
  26. 26.How many factors does 60 have? (a) 13 (b) 11 (c) 12 (d) 10
  27. 27.How many factors does 117 have? (a) 6 (b) 7 (c) 4 (d) 5
  28. 28.How many factors does 23 have? (a) 2 (b) 3 (c) 4 (d) 1
  29. 29.How many prime numbers are there between 90 and 110? (a) 6 (b) 7 (c) 4 (d) 5
  30. 30.What is the smallest prime factor of 155? (a) 2 (b) 5 (c) 31 (d) 1
  31. 31.What is the prime factorisation of 63? (a) 3² × 7 (b) 3³ × 7 (c) 3 × 7² (d) 3 × 7
  32. 32.How many factors does 47 have? (a) 4 (b) 1 (c) 2 (d) 3
  33. 33.What is the prime factorisation of 36? (a) 2 × 3² (b) 2³ × 3² (c) 2² × 3³ (d) 2² × 3²
  34. 34.How many factors does 119 have? (a) 4 (b) 3 (c) 2 (d) 5
  35. 35.What is the prime factorisation of 12? (a) 2 × 3 (b) 2 × 3² (c) 2² × 3 (d) 2³ × 3
  36. 36.How many factors does 67 have? (a) 3 (b) 1 (c) 2 (d) 4
  37. 37.What is the prime factorisation of 18? (a) 2 × 3³ (b) 2 × 3² (c) 2² × 3 (d) 2² × 3²
  38. 38.What is the smallest prime factor of 33? (a) 2 (b) 1 (c) 11 (d) 3
  39. 39.What is the prime factorisation of 90? (a) 2³ × 3² × 5 (b) 2 × 3² × 5 (c) 2² × 3² × 5 (d) 2 × 3² × 5²
  40. 40.What is the smallest prime factor of 55? (a) 1 (b) 11 (c) 5 (d) 2

Answer key

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

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