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Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 86 · 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.How many prime numbers are there between 70 and 90? (a) 4 (b) 6 (c) 7 (d) 5
  2. 2.What is the prime factorisation of 15? (a) 3² × 5 (b) 3³ × 5 (c) 3 × 5 (d) 3 × 5²
  3. 3.What is the prime factorisation of 126? (a) 2² × 3² × 7 (b) 2 × 3² × 7 (c) 2 × 3² × 7² (d) 2³ × 3² × 7
  4. 4.How many factors does 12 have? (a) 4 (b) 5 (c) 6 (d) 7
  5. 5.How many prime numbers are there between 20 and 30? (a) 4 (b) 3 (c) 1 (d) 2
  6. 6.What is the smallest prime factor of 341? (a) 11 (b) 1 (c) 31 (d) 2
  7. 7.How many factors does 66 have? (a) 7 (b) 6 (c) 9 (d) 8
  8. 8.What is the smallest prime factor of 217? (a) 7 (b) 1 (c) 2 (d) 31
  9. 9.What is the prime factorisation of 48? (a) 2⁵ × 3 (b) 2⁴ × 3 (c) 2³ × 3 (d) 2 × 3⁴
  10. 10.What is the prime factorisation of 30? (a) 2² × 3 × 5 (b) 2 × 3 × 5² (c) 2³ × 3 × 5 (d) 2 × 3 × 5
  11. 11.What is the smallest prime factor of 121? (a) 1 (b) 11 (c) 3 (d) 2
  12. 12.What is the smallest prime factor of 85? (a) 5 (b) 17 (c) 2 (d) 1
  13. 13.How many factors does 57 have? (a) 3 (b) 4 (c) 2 (d) 5
  14. 14.What is the prime factorisation of 360? (a) 2 × 3² × 5³ (b) 2² × 3² × 5 (c) 2⁴ × 3² × 5 (d) 2³ × 3² × 5
  15. 15.What is the prime factorisation of 12? (a) 2² × 3 (b) 2 × 3 (c) 2 × 3² (d) 2³ × 3
  16. 16.How many prime numbers are there between 10 and 30? (a) 6 (b) 5 (c) 7 (d) 8
  17. 17.How many factors does 54 have? (a) 9 (b) 6 (c) 8 (d) 7
  18. 18.What is the prime factorisation of 105? (a) 3³ × 5 × 7 (b) 3² × 5 × 7 (c) 3 × 5 × 7² (d) 3 × 5 × 7
  19. 19.How many factors does 70 have? (a) 7 (b) 6 (c) 8 (d) 9
  20. 20.What is the prime factorisation of 144? (a) 2⁵ × 3² (b) 2² × 3⁴ (c) 2⁴ × 3² (d) 2³ × 3²
  21. 21.What is the smallest prime factor of 209? (a) 2 (b) 19 (c) 11 (d) 1
  22. 22.How many factors does 75 have? (a) 6 (b) 7 (c) 4 (d) 5
  23. 23.How many factors does 115 have? (a) 2 (b) 4 (c) 5 (d) 3
  24. 24.How many factors does 120 have? (a) 14 (b) 16 (c) 17 (d) 15
  25. 25.What is the smallest prime factor of 155? (a) 31 (b) 2 (c) 5 (d) 1
  26. 26.How many factors does 45 have? (a) 4 (b) 6 (c) 5 (d) 7
  27. 27.What is the prime factorisation of 140? (a) 2 × 5 × 7 (b) 2 × 5 × 7² (c) 2³ × 5 × 7 (d) 2² × 5 × 7
  28. 28.What is the prime factorisation of 63? (a) 3 × 7 (b) 3 × 7² (c) 3³ × 7 (d) 3² × 7
  29. 29.What is the prime factorisation of 168? (a) 2⁴ × 3 × 7 (b) 2 × 3 × 7³ (c) 2³ × 3 × 7 (d) 2² × 3 × 7
  30. 30.What is the prime factorisation of 80? (a) 2⁵ × 5 (b) 2 × 5⁴ (c) 2³ × 5 (d) 2⁴ × 5
  31. 31.How many prime numbers are there between 70 and 80? (a) 2 (b) 4 (c) 3 (d) 5
  32. 32.What is the prime factorisation of 56? (a) 2 × 7³ (b) 2² × 7 (c) 2⁴ × 7 (d) 2³ × 7
  33. 33.What is the smallest prime factor of 21? (a) 2 (b) 1 (c) 3 (d) 7
  34. 34.What is the smallest prime factor of 161? (a) 23 (b) 1 (c) 7 (d) 2
  35. 35.What is the prime factorisation of 18? (a) 2 × 3³ (b) 2² × 3 (c) 2² × 3² (d) 2 × 3²
  36. 36.How many factors does 87 have? (a) 3 (b) 5 (c) 2 (d) 4
  37. 37.What is the smallest prime factor of 87? (a) 3 (b) 29 (c) 1 (d) 2
  38. 38.What is the prime factorisation of 36? (a) 2² × 3³ (b) 2² × 3² (c) 2³ × 3² (d) 2 × 3²
  39. 39.How many prime numbers are there between 90 and 110? (a) 4 (b) 5 (c) 6 (d) 7
  40. 40.How many factors does 22 have? (a) 3 (b) 2 (c) 5 (d) 4

Answer key

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

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