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Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 30 · 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 20 and 50? (a) 7 (b) 8 (c) 6 (d) 9
  2. 2.What is the prime factorisation of 144? (a) 2³ × 3² (b) 2² × 3⁴ (c) 2⁴ × 3² (d) 2⁵ × 3²
  3. 3.How many factors does 72 have? (a) 10 (b) 11 (c) 13 (d) 12
  4. 4.What is the prime factorisation of 14? (a) 2³ × 7 (b) 2 × 7 (c) 2 × 7² (d) 2² × 7
  5. 5.How many factors does 43 have? (a) 3 (b) 2 (c) 4 (d) 1
  6. 6.How many factors does 24 have? (a) 6 (b) 9 (c) 8 (d) 7
  7. 7.How many prime numbers are there between 70 and 80? (a) 4 (b) 2 (c) 3 (d) 5
  8. 8.What is the prime factorisation of 70? (a) 2 × 5 × 7 (b) 2 × 5 × 7² (c) 2³ × 5 × 7 (d) 2² × 5 × 7
  9. 9.How many prime numbers are there between 90 and 110? (a) 7 (b) 6 (c) 4 (d) 5
  10. 10.How many prime numbers are there between 20 and 30? (a) 3 (b) 1 (c) 2 (d) 4
  11. 11.What is the smallest prime factor of 145? (a) 2 (b) 1 (c) 29 (d) 5
  12. 12.How many prime numbers are there between 30 and 50? (a) 4 (b) 5 (c) 6 (d) 7
  13. 13.What is the smallest prime factor of 133? (a) 2 (b) 19 (c) 7 (d) 1
  14. 14.How many factors does 30 have? (a) 7 (b) 8 (c) 6 (d) 9
  15. 15.What is the prime factorisation of 30? (a) 2 × 3 × 5² (b) 2² × 3 × 5 (c) 2³ × 3 × 5 (d) 2 × 3 × 5
  16. 16.How many factors does 120 have? (a) 16 (b) 15 (c) 14 (d) 17
  17. 17.How many factors does 37 have? (a) 1 (b) 3 (c) 4 (d) 2
  18. 18.What is the prime factorisation of 280? (a) 2⁴ × 5 × 7 (b) 2² × 5 × 7 (c) 2³ × 5 × 7 (d) 2 × 5 × 7³
  19. 19.What is the prime factorisation of 90? (a) 2² × 3² × 5 (b) 2 × 3² × 5 (c) 2 × 3² × 5² (d) 2³ × 3² × 5
  20. 20.What is the smallest prime factor of 209? (a) 1 (b) 2 (c) 11 (d) 19
  21. 21.How many factors does 77 have? (a) 3 (b) 5 (c) 4 (d) 2
  22. 22.How many prime numbers are there between 30 and 60? (a) 9 (b) 8 (c) 7 (d) 6
  23. 23.How many prime numbers are there between 50 and 70? (a) 5 (b) 3 (c) 6 (d) 4
  24. 24.How many prime numbers are there between 50 and 60? (a) 3 (b) 4 (c) 1 (d) 2
  25. 25.How many factors does 100 have? (a) 9 (b) 7 (c) 10 (d) 8
  26. 26.How many factors does 89 have? (a) 2 (b) 3 (c) 1 (d) 4
  27. 27.How many factors does 95 have? (a) 2 (b) 5 (c) 3 (d) 4
  28. 28.How many factors does 45 have? (a) 6 (b) 7 (c) 5 (d) 4
  29. 29.What is the prime factorisation of 56? (a) 2² × 7 (b) 2³ × 7 (c) 2⁴ × 7 (d) 2 × 7³
  30. 30.What is the prime factorisation of 60? (a) 2³ × 3 × 5 (b) 2² × 3 × 5 (c) 2 × 3 × 5² (d) 2 × 3 × 5
  31. 31.What is the prime factorisation of 36? (a) 2² × 3³ (b) 2 × 3² (c) 2² × 3² (d) 2³ × 3²
  32. 32.How many factors does 114 have? (a) 8 (b) 6 (c) 7 (d) 9
  33. 33.What is the prime factorisation of 336? (a) 2 × 3 × 7⁴ (b) 2³ × 3 × 7 (c) 2⁴ × 3 × 7 (d) 2⁵ × 3 × 7
  34. 34.How many factors does 73 have? (a) 4 (b) 1 (c) 3 (d) 2
  35. 35.How many prime numbers are there between 90 and 120? (a) 5 (b) 8 (c) 6 (d) 7
  36. 36.How many factors does 32 have? (a) 5 (b) 7 (c) 4 (d) 6
  37. 37.What is the smallest prime factor of 253? (a) 23 (b) 11 (c) 2 (d) 1
  38. 38.How many factors does 18 have? (a) 4 (b) 6 (c) 7 (d) 5
  39. 39.How many factors does 13 have? (a) 1 (b) 2 (c) 4 (d) 3
  40. 40.What is the smallest prime factor of 55? (a) 11 (b) 2 (c) 1 (d) 5

Answer key

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

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