Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 52 · 40 questions

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.

1. 1.How many factors does 79 have? (a) 3 (b) 2 (c) 1 (d) 4
2. 2.How many factors does 102 have? (a) 7 (b) 6 (c) 8 (d) 9
3. 3.What is the prime factorisation of 15? (a) 3 × 5² (b) 3² × 5 (c) 3 × 5 (d) 3³ × 5
4. 4.How many factors does 69 have? (a) 4 (b) 3 (c) 5 (d) 2
5. 5.What is the prime factorisation of 140? (a) 2 × 5 × 7² (b) 2 × 5 × 7 (c) 2² × 5 × 7 (d) 2³ × 5 × 7
6. 6.What is the prime factorisation of 560? (a) 2⁴ × 5 × 7 (b) 2⁵ × 5 × 7 (c) 2 × 5 × 7⁴ (d) 2³ × 5 × 7
7. 7.How many prime numbers are there between 30 and 40? (a) 1 (b) 3 (c) 4 (d) 2
8. 8.What is the smallest prime factor of 39? (a) 2 (b) 13 (c) 1 (d) 3
9. 9.How many factors does 22 have? (a) 3 (b) 5 (c) 4 (d) 2
10. 10.How many factors does 37 have? (a) 3 (b) 4 (c) 1 (d) 2
11. 11.What is the prime factorisation of 84? (a) 2³ × 3 × 7 (b) 2² × 3 × 7 (c) 2 × 3 × 7² (d) 2 × 3 × 7
12. 12.How many factors does 99 have? (a) 6 (b) 5 (c) 4 (d) 7
13. 13.How many factors does 118 have? (a) 3 (b) 5 (c) 2 (d) 4
14. 14.What is the smallest prime factor of 65? (a) 1 (b) 5 (c) 2 (d) 13
15. 15.What is the prime factorisation of 12? (a) 2 × 3² (b) 2³ × 3 (c) 2 × 3 (d) 2² × 3
16. 16.How many factors does 92 have? (a) 5 (b) 7 (c) 6 (d) 4
17. 17.What is the prime factorisation of 70? (a) 2 × 5 × 7 (b) 2² × 5 × 7 (c) 2 × 5 × 7² (d) 2³ × 5 × 7
18. 18.How many factors does 46 have? (a) 4 (b) 2 (c) 3 (d) 5
19. 19.How many factors does 25 have? (a) 2 (b) 4 (c) 3 (d) 1
20. 20.How many prime numbers are there between 80 and 90? (a) 4 (b) 1 (c) 2 (d) 3
21. 21.What is the prime factorisation of 21? (a) 3³ × 7 (b) 3 × 7 (c) 3² × 7 (d) 3 × 7²
22. 22.What is the prime factorisation of 28? (a) 2 × 7 (b) 2³ × 7 (c) 2 × 7² (d) 2² × 7
23. 23.What is the prime factorisation of 105? (a) 3 × 5 × 7² (b) 3 × 5 × 7 (c) 3³ × 5 × 7 (d) 3² × 5 × 7
24. 24.How many factors does 63 have? (a) 6 (b) 4 (c) 7 (d) 5
25. 25.What is the smallest prime factor of 145? (a) 2 (b) 29 (c) 5 (d) 1
26. 26.What is the smallest prime factor of 87? (a) 3 (b) 29 (c) 1 (d) 2
27. 27.How many factors does 71 have? (a) 3 (b) 1 (c) 4 (d) 2
28. 28.What is the prime factorisation of 56? (a) 2² × 7 (b) 2⁴ × 7 (c) 2 × 7³ (d) 2³ × 7
29. 29.How many factors does 91 have? (a) 5 (b) 3 (c) 4 (d) 2
30. 30.What is the smallest prime factor of 55? (a) 11 (b) 2 (c) 5 (d) 1
31. 31.How many prime numbers are there between 80 and 110? (a) 8 (b) 7 (c) 6 (d) 9
32. 32.How many factors does 59 have? (a) 1 (b) 4 (c) 3 (d) 2
33. 33.How many factors does 70 have? (a) 7 (b) 6 (c) 9 (d) 8
34. 34.What is the smallest prime factor of 221? (a) 13 (b) 17 (c) 1 (d) 2
35. 35.What is the prime factorisation of 420? (a) 2³ × 3 × 5 × 7 (b) 2 × 3 × 5 × 7 (c) 2 × 3 × 5 × 7² (d) 2² × 3 × 5 × 7
36. 36.What is the smallest prime factor of 49? (a) 7 (b) 2 (c) 3 (d) 1
37. 37.How many factors does 105 have? (a) 6 (b) 9 (c) 7 (d) 8
38. 38.What is the prime factorisation of 180? (a) 2 × 3² × 5 (b) 2 × 3² × 5² (c) 2³ × 3² × 5 (d) 2² × 3² × 5
39. 39.What is the prime factorisation of 360? (a) 2⁴ × 3² × 5 (b) 2 × 3² × 5³ (c) 2³ × 3² × 5 (d) 2² × 3² × 5
40. 40.How many prime numbers are there between 60 and 70? (a) 2 (b) 4 (c) 3 (d) 1

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