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Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 118 · 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 115? (a) 1 (b) 5 (c) 23 (d) 2
  2. 2.How many factors does 111 have? (a) 3 (b) 4 (c) 5 (d) 2
  3. 3.How many prime numbers are there between 60 and 70? (a) 2 (b) 3 (c) 1 (d) 4
  4. 4.What is the prime factorisation of 360? (a) 2 × 3² × 5³ (b) 2⁴ × 3² × 5 (c) 2² × 3² × 5 (d) 2³ × 3² × 5
  5. 5.What is the smallest prime factor of 169? (a) 2 (b) 3 (c) 13 (d) 1
  6. 6.What is the prime factorisation of 90? (a) 2 × 3² × 5² (b) 2 × 3² × 5 (c) 2³ × 3² × 5 (d) 2² × 3² × 5
  7. 7.How many factors does 105 have? (a) 6 (b) 7 (c) 9 (d) 8
  8. 8.How many factors does 84 have? (a) 11 (b) 10 (c) 13 (d) 12
  9. 9.How many factors does 74 have? (a) 3 (b) 5 (c) 2 (d) 4
  10. 10.How many factors does 110 have? (a) 8 (b) 7 (c) 9 (d) 6
  11. 11.What is the smallest prime factor of 155? (a) 31 (b) 2 (c) 5 (d) 1
  12. 12.How many factors does 17 have? (a) 3 (b) 1 (c) 4 (d) 2
  13. 13.What is the smallest prime factor of 121? (a) 1 (b) 11 (c) 2 (d) 3
  14. 14.How many factors does 57 have? (a) 4 (b) 2 (c) 3 (d) 5
  15. 15.How many prime numbers are there between 20 and 50? (a) 9 (b) 6 (c) 7 (d) 8
  16. 16.How many factors does 92 have? (a) 4 (b) 6 (c) 5 (d) 7
  17. 17.How many prime numbers are there between 40 and 60? (a) 7 (b) 4 (c) 5 (d) 6
  18. 18.How many factors does 36 have? (a) 7 (b) 9 (c) 10 (d) 8
  19. 19.What is the prime factorisation of 210? (a) 2³ × 3 × 5 × 7 (b) 2 × 3 × 5 × 7 (c) 2² × 3 × 5 × 7 (d) 2 × 3 × 5 × 7²
  20. 20.What is the prime factorisation of 60? (a) 2³ × 3 × 5 (b) 2 × 3 × 5² (c) 2² × 3 × 5 (d) 2 × 3 × 5
  21. 21.How many factors does 104 have? (a) 6 (b) 9 (c) 8 (d) 7
  22. 22.How many prime numbers are there between 30 and 50? (a) 6 (b) 5 (c) 4 (d) 7
  23. 23.How many prime numbers are there between 50 and 60? (a) 4 (b) 3 (c) 1 (d) 2
  24. 24.What is the smallest prime factor of 209? (a) 11 (b) 2 (c) 19 (d) 1
  25. 25.How many factors does 13 have? (a) 4 (b) 1 (c) 3 (d) 2
  26. 26.How many prime numbers are there between 90 and 100? (a) 0 (b) 2 (c) 3 (d) 1
  27. 27.How many prime numbers are there between 70 and 80? (a) 5 (b) 2 (c) 4 (d) 3
  28. 28.How many factors does 91 have? (a) 2 (b) 5 (c) 3 (d) 4
  29. 29.How many factors does 113 have? (a) 3 (b) 2 (c) 4 (d) 1
  30. 30.How many factors does 42 have? (a) 9 (b) 7 (c) 8 (d) 6
  31. 31.What is the smallest prime factor of 85? (a) 1 (b) 5 (c) 17 (d) 2
  32. 32.What is the smallest prime factor of 33? (a) 1 (b) 11 (c) 2 (d) 3
  33. 33.How many factors does 86 have? (a) 3 (b) 4 (c) 5 (d) 2
  34. 34.How many prime numbers are there between 90 and 110? (a) 7 (b) 4 (c) 5 (d) 6
  35. 35.What is the prime factorisation of 144? (a) 2² × 3⁴ (b) 2³ × 3² (c) 2⁵ × 3² (d) 2⁴ × 3²
  36. 36.What is the smallest prime factor of 377? (a) 1 (b) 29 (c) 13 (d) 2
  37. 37.What is the prime factorisation of 70? (a) 2³ × 5 × 7 (b) 2² × 5 × 7 (c) 2 × 5 × 7² (d) 2 × 5 × 7
  38. 38.What is the prime factorisation of 21? (a) 3 × 7² (b) 3² × 7 (c) 3 × 7 (d) 3³ × 7
  39. 39.What is the prime factorisation of 112? (a) 2⁵ × 7 (b) 2 × 7⁴ (c) 2⁴ × 7 (d) 2³ × 7
  40. 40.What is the prime factorisation of 84? (a) 2 × 3 × 7² (b) 2 × 3 × 7 (c) 2³ × 3 × 7 (d) 2² × 3 × 7

Answer key

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

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