Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 80 · 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 prime numbers are there between 20 and 50? (a) 6 (b) 7 (c) 8 (d) 9
2. 2.How many factors does 31 have? (a) 2 (b) 4 (c) 1 (d) 3
3. 3.What is the smallest prime factor of 377? (a) 13 (b) 1 (c) 2 (d) 29
4. 4.What is the smallest prime factor of 77? (a) 1 (b) 7 (c) 11 (d) 2
5. 5.How many factors does 29 have? (a) 1 (b) 4 (c) 3 (d) 2
6. 6.What is the smallest prime factor of 161? (a) 1 (b) 23 (c) 2 (d) 7
7. 7.How many factors does 79 have? (a) 1 (b) 4 (c) 3 (d) 2
8. 8.How many prime numbers are there between 20 and 30? (a) 1 (b) 2 (c) 4 (d) 3
9. 9.What is the prime factorisation of 420? (a) 2 × 3 × 5 × 7 (b) 2³ × 3 × 5 × 7 (c) 2 × 3 × 5 × 7² (d) 2² × 3 × 5 × 7
10. 10.How many prime numbers are there between 30 and 40? (a) 4 (b) 1 (c) 3 (d) 2
11. 11.What is the smallest prime factor of 121? (a) 1 (b) 2 (c) 3 (d) 11
12. 12.How many factors does 107 have? (a) 3 (b) 1 (c) 2 (d) 4
13. 13.How many prime numbers are there between 50 and 80? (a) 6 (b) 9 (c) 7 (d) 8
14. 14.What is the smallest prime factor of 85? (a) 17 (b) 2 (c) 1 (d) 5
15. 15.What is the smallest prime factor of 145? (a) 5 (b) 1 (c) 2 (d) 29
16. 16.What is the prime factorisation of 40? (a) 2 × 5³ (b) 2³ × 5 (c) 2⁴ × 5 (d) 2² × 5
17. 17.What is the prime factorisation of 36? (a) 2² × 3³ (b) 2² × 3² (c) 2 × 3² (d) 2³ × 3²
18. 18.How many factors does 90 have? (a) 10 (b) 12 (c) 11 (d) 13
19. 19.How many factors does 38 have? (a) 3 (b) 2 (c) 5 (d) 4
20. 20.How many prime numbers are there between 90 and 110? (a) 4 (b) 5 (c) 7 (d) 6
21. 21.What is the prime factorisation of 12? (a) 2 × 3² (b) 2² × 3 (c) 2 × 3 (d) 2³ × 3
22. 22.What is the prime factorisation of 45? (a) 3 × 5 (b) 3² × 5 (c) 3 × 5² (d) 3³ × 5
23. 23.How many factors does 30 have? (a) 7 (b) 9 (c) 6 (d) 8
24. 24.How many prime numbers are there between 10 and 30? (a) 7 (b) 5 (c) 8 (d) 6
25. 25.How many factors does 37 have? (a) 2 (b) 4 (c) 1 (d) 3
26. 26.What is the prime factorisation of 63? (a) 3 × 7² (b) 3³ × 7 (c) 3 × 7 (d) 3² × 7
27. 27.What is the prime factorisation of 30? (a) 2² × 3 × 5 (b) 2 × 3 × 5 (c) 2 × 3 × 5² (d) 2³ × 3 × 5
28. 28.What is the prime factorisation of 42? (a) 2³ × 3 × 7 (b) 2² × 3 × 7 (c) 2 × 3 × 7² (d) 2 × 3 × 7
29. 29.What is the prime factorisation of 84? (a) 2³ × 3 × 7 (b) 2 × 3 × 7² (c) 2² × 3 × 7 (d) 2 × 3 × 7
30. 30.How many factors does 120 have? (a) 17 (b) 16 (c) 15 (d) 14
31. 31.What is the prime factorisation of 315? (a) 3 × 5 × 7 (b) 3 × 5 × 7² (c) 3³ × 5 × 7 (d) 3² × 5 × 7
32. 32.How many factors does 36 have? (a) 10 (b) 9 (c) 7 (d) 8
33. 33.How many prime numbers are there between 50 and 60? (a) 1 (b) 3 (c) 2 (d) 4
34. 34.How many factors does 102 have? (a) 6 (b) 7 (c) 9 (d) 8
35. 35.What is the prime factorisation of 56? (a) 2⁴ × 7 (b) 2³ × 7 (c) 2² × 7 (d) 2 × 7³
36. 36.How many factors does 73 have? (a) 3 (b) 4 (c) 2 (d) 1
37. 37.How many prime numbers are there between 10 and 20? (a) 5 (b) 3 (c) 6 (d) 4
38. 38.What is the prime factorisation of 560? (a) 2⁴ × 5 × 7 (b) 2⁵ × 5 × 7 (c) 2³ × 5 × 7 (d) 2 × 5 × 7⁴
39. 39.How many prime numbers are there between 40 and 50? (a) 3 (b) 5 (c) 4 (d) 2
40. 40.What is the prime factorisation of 21? (a) 3 × 7 (b) 3³ × 7 (c) 3² × 7 (d) 3 × 7²

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