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Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 171 · 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 112? (a) 2³ × 7 (b) 2⁵ × 7 (c) 2 × 7⁴ (d) 2⁴ × 7
  2. 2.How many prime numbers are there between 80 and 110? (a) 6 (b) 9 (c) 8 (d) 7
  3. 3.How many prime numbers are there between 50 and 70? (a) 3 (b) 4 (c) 6 (d) 5
  4. 4.How many factors does 109 have? (a) 3 (b) 1 (c) 4 (d) 2
  5. 5.What is the prime factorisation of 210? (a) 2 × 3 × 5 × 7² (b) 2 × 3 × 5 × 7 (c) 2² × 3 × 5 × 7 (d) 2³ × 3 × 5 × 7
  6. 6.What is the smallest prime factor of 143? (a) 13 (b) 11 (c) 2 (d) 1
  7. 7.How many factors does 106 have? (a) 5 (b) 2 (c) 3 (d) 4
  8. 8.What is the prime factorisation of 60? (a) 2 × 3 × 5 (b) 2 × 3 × 5² (c) 2³ × 3 × 5 (d) 2² × 3 × 5
  9. 9.How many factors does 84 have? (a) 11 (b) 10 (c) 12 (d) 13
  10. 10.What is the prime factorisation of 40? (a) 2 × 5³ (b) 2⁴ × 5 (c) 2² × 5 (d) 2³ × 5
  11. 11.What is the smallest prime factor of 51? (a) 17 (b) 3 (c) 2 (d) 1
  12. 12.How many factors does 51 have? (a) 3 (b) 2 (c) 5 (d) 4
  13. 13.What is the smallest prime factor of 217? (a) 1 (b) 7 (c) 2 (d) 31
  14. 14.How many factors does 101 have? (a) 1 (b) 4 (c) 3 (d) 2
  15. 15.What is the prime factorisation of 28? (a) 2³ × 7 (b) 2² × 7 (c) 2 × 7 (d) 2 × 7²
  16. 16.How many factors does 46 have? (a) 2 (b) 3 (c) 4 (d) 5
  17. 17.What is the prime factorisation of 504? (a) 2⁴ × 3² × 7 (b) 2 × 3² × 7³ (c) 2² × 3² × 7 (d) 2³ × 3² × 7
  18. 18.How many prime numbers are there between 40 and 70? (a) 6 (b) 9 (c) 8 (d) 7
  19. 19.What is the smallest prime factor of 35? (a) 1 (b) 7 (c) 2 (d) 5
  20. 20.How many prime numbers are there between 10 and 40? (a) 9 (b) 8 (c) 10 (d) 7
  21. 21.What is the smallest prime factor of 155? (a) 5 (b) 1 (c) 2 (d) 31
  22. 22.How many factors does 108 have? (a) 11 (b) 13 (c) 10 (d) 12
  23. 23.How many prime numbers are there between 30 and 40? (a) 1 (b) 3 (c) 2 (d) 4
  24. 24.What is the prime factorisation of 120? (a) 2² × 3 × 5 (b) 2⁴ × 3 × 5 (c) 2 × 3 × 5³ (d) 2³ × 3 × 5
  25. 25.What is the smallest prime factor of 115? (a) 1 (b) 23 (c) 2 (d) 5
  26. 26.How many factors does 36 have? (a) 10 (b) 8 (c) 9 (d) 7
  27. 27.How many prime numbers are there between 20 and 40? (a) 3 (b) 5 (c) 6 (d) 4
  28. 28.How many factors does 14 have? (a) 5 (b) 4 (c) 3 (d) 2
  29. 29.How many factors does 111 have? (a) 2 (b) 5 (c) 3 (d) 4
  30. 30.What is the prime factorisation of 20? (a) 2 × 5 (b) 2³ × 5 (c) 2² × 5 (d) 2 × 5²
  31. 31.How many prime numbers are there between 90 and 120? (a) 8 (b) 6 (c) 7 (d) 5
  32. 32.What is the prime factorisation of 18? (a) 2 × 3² (b) 2 × 3³ (c) 2² × 3 (d) 2² × 3²
  33. 33.How many prime numbers are there between 50 and 80? (a) 8 (b) 6 (c) 9 (d) 7
  34. 34.How many factors does 33 have? (a) 5 (b) 4 (c) 2 (d) 3
  35. 35.How many factors does 71 have? (a) 4 (b) 1 (c) 3 (d) 2
  36. 36.What is the smallest prime factor of 221? (a) 2 (b) 1 (c) 13 (d) 17
  37. 37.How many factors does 114 have? (a) 8 (b) 6 (c) 7 (d) 9
  38. 38.How many factors does 23 have? (a) 3 (b) 4 (c) 1 (d) 2
  39. 39.What is the smallest prime factor of 85? (a) 1 (b) 17 (c) 5 (d) 2
  40. 40.What is the prime factorisation of 63? (a) 3 × 7² (b) 3² × 7 (c) 3³ × 7 (d) 3 × 7

Answer key

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

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