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Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 193 · 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 factors does 33 have? (a) 3 (b) 5 (c) 2 (d) 4
  2. 2.How many factors does 106 have? (a) 3 (b) 2 (c) 4 (d) 5
  3. 3.What is the prime factorisation of 144? (a) 2⁴ × 3² (b) 2³ × 3² (c) 2⁵ × 3² (d) 2² × 3⁴
  4. 4.What is the smallest prime factor of 69? (a) 3 (b) 23 (c) 2 (d) 1
  5. 5.How many factors does 44 have? (a) 6 (b) 5 (c) 7 (d) 4
  6. 6.How many factors does 111 have? (a) 3 (b) 2 (c) 5 (d) 4
  7. 7.How many prime numbers are there between 90 and 110? (a) 7 (b) 4 (c) 6 (d) 5
  8. 8.What is the prime factorisation of 35? (a) 5² × 7 (b) 5 × 7² (c) 5³ × 7 (d) 5 × 7
  9. 9.How many factors does 89 have? (a) 1 (b) 2 (c) 4 (d) 3
  10. 10.What is the smallest prime factor of 35? (a) 7 (b) 2 (c) 5 (d) 1
  11. 11.How many factors does 82 have? (a) 3 (b) 2 (c) 5 (d) 4
  12. 12.What is the prime factorisation of 21? (a) 3² × 7 (b) 3 × 7² (c) 3 × 7 (d) 3³ × 7
  13. 13.What is the smallest prime factor of 247? (a) 19 (b) 1 (c) 13 (d) 2
  14. 14.What is the smallest prime factor of 91? (a) 7 (b) 13 (c) 1 (d) 2
  15. 15.How many prime numbers are there between 80 and 110? (a) 9 (b) 8 (c) 7 (d) 6
  16. 16.How many factors does 99 have? (a) 7 (b) 5 (c) 4 (d) 6
  17. 17.How many factors does 97 have? (a) 1 (b) 2 (c) 4 (d) 3
  18. 18.What is the prime factorisation of 240? (a) 2⁵ × 3 × 5 (b) 2³ × 3 × 5 (c) 2 × 3 × 5⁴ (d) 2⁴ × 3 × 5
  19. 19.What is the prime factorisation of 80? (a) 2 × 5⁴ (b) 2⁵ × 5 (c) 2³ × 5 (d) 2⁴ × 5
  20. 20.What is the smallest prime factor of 203? (a) 29 (b) 1 (c) 2 (d) 7
  21. 21.What is the smallest prime factor of 209? (a) 2 (b) 19 (c) 11 (d) 1
  22. 22.How many factors does 41 have? (a) 4 (b) 1 (c) 3 (d) 2
  23. 23.How many prime numbers are there between 30 and 50? (a) 7 (b) 5 (c) 6 (d) 4
  24. 24.What is the prime factorisation of 12? (a) 2³ × 3 (b) 2 × 3² (c) 2² × 3 (d) 2 × 3
  25. 25.How many prime numbers are there between 40 and 70? (a) 9 (b) 8 (c) 7 (d) 6
  26. 26.How many prime numbers are there between 40 and 50? (a) 4 (b) 5 (c) 3 (d) 2
  27. 27.How many factors does 47 have? (a) 1 (b) 4 (c) 2 (d) 3
  28. 28.What is the prime factorisation of 560? (a) 2⁵ × 5 × 7 (b) 2⁴ × 5 × 7 (c) 2 × 5 × 7⁴ (d) 2³ × 5 × 7
  29. 29.What is the smallest prime factor of 33? (a) 1 (b) 11 (c) 3 (d) 2
  30. 30.How many factors does 74 have? (a) 3 (b) 2 (c) 5 (d) 4
  31. 31.How many prime numbers are there between 60 and 70? (a) 1 (b) 3 (c) 2 (d) 4
  32. 32.How many factors does 15 have? (a) 4 (b) 2 (c) 5 (d) 3
  33. 33.What is the smallest prime factor of 21? (a) 1 (b) 7 (c) 2 (d) 3
  34. 34.What is the prime factorisation of 105? (a) 3 × 5 × 7 (b) 3 × 5 × 7² (c) 3³ × 5 × 7 (d) 3² × 5 × 7
  35. 35.What is the smallest prime factor of 145? (a) 29 (b) 1 (c) 2 (d) 5
  36. 36.How many prime numbers are there between 20 and 30? (a) 1 (b) 3 (c) 2 (d) 4
  37. 37.How many prime numbers are there between 10 and 20? (a) 4 (b) 5 (c) 3 (d) 6
  38. 38.How many factors does 19 have? (a) 1 (b) 3 (c) 4 (d) 2
  39. 39.How many factors does 22 have? (a) 4 (b) 5 (c) 2 (d) 3
  40. 40.How many factors does 67 have? (a) 4 (b) 1 (c) 2 (d) 3

Answer key

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

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