Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 11 · 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.What is the prime factorisation of 35? (a) 5³ × 7 (b) 5 × 7 (c) 5² × 7 (d) 5 × 7²
2. 2.What is the prime factorisation of 126? (a) 2 × 3² × 7 (b) 2³ × 3² × 7 (c) 2 × 3² × 7² (d) 2² × 3² × 7
3. 3.How many factors does 23 have? (a) 3 (b) 4 (c) 1 (d) 2
4. 4.How many prime numbers are there between 80 and 100? (a) 5 (b) 2 (c) 4 (d) 3
5. 5.How many factors does 50 have? (a) 4 (b) 5 (c) 6 (d) 7
6. 6.How many factors does 77 have? (a) 3 (b) 5 (c) 2 (d) 4
7. 7.How many factors does 67 have? (a) 4 (b) 1 (c) 3 (d) 2
8. 8.How many prime numbers are there between 40 and 70? (a) 6 (b) 8 (c) 9 (d) 7
9. 9.How many factors does 32 have? (a) 5 (b) 6 (c) 7 (d) 4
10. 10.What is the prime factorisation of 45? (a) 3³ × 5 (b) 3 × 5 (c) 3 × 5² (d) 3² × 5
11. 11.What is the prime factorisation of 15? (a) 3 × 5² (b) 3² × 5 (c) 3³ × 5 (d) 3 × 5
12. 12.How many prime numbers are there between 30 and 50? (a) 5 (b) 6 (c) 4 (d) 7
13. 13.What is the prime factorisation of 70? (a) 2 × 5 × 7 (b) 2 × 5 × 7² (c) 2³ × 5 × 7 (d) 2² × 5 × 7
14. 14.What is the smallest prime factor of 93? (a) 31 (b) 1 (c) 3 (d) 2
15. 15.What is the prime factorisation of 105? (a) 3 × 5 × 7² (b) 3² × 5 × 7 (c) 3 × 5 × 7 (d) 3³ × 5 × 7
16. 16.What is the prime factorisation of 21? (a) 3 × 7 (b) 3³ × 7 (c) 3 × 7² (d) 3² × 7
17. 17.How many prime numbers are there between 90 and 100? (a) 2 (b) 1 (c) 3 (d) 0
18. 18.How many prime numbers are there between 60 and 70? (a) 1 (b) 4 (c) 2 (d) 3
19. 19.How many factors does 42 have? (a) 6 (b) 9 (c) 8 (d) 7
20. 20.What is the prime factorisation of 63? (a) 3 × 7 (b) 3³ × 7 (c) 3² × 7 (d) 3 × 7²
21. 21.What is the prime factorisation of 18? (a) 2 × 3² (b) 2² × 3² (c) 2² × 3 (d) 2 × 3³
22. 22.How many prime numbers are there between 50 and 60? (a) 1 (b) 3 (c) 4 (d) 2
23. 23.What is the prime factorisation of 56? (a) 2⁴ × 7 (b) 2³ × 7 (c) 2 × 7³ (d) 2² × 7
24. 24.How many prime numbers are there between 40 and 50? (a) 2 (b) 3 (c) 5 (d) 4
25. 25.What is the smallest prime factor of 161? (a) 1 (b) 7 (c) 23 (d) 2
26. 26.How many prime numbers are there between 20 and 40? (a) 5 (b) 3 (c) 6 (d) 4
27. 27.What is the smallest prime factor of 69? (a) 1 (b) 23 (c) 3 (d) 2
28. 28.How many prime numbers are there between 10 and 30? (a) 5 (b) 7 (c) 8 (d) 6
29. 29.How many factors does 104 have? (a) 8 (b) 6 (c) 9 (d) 7
30. 30.How many prime numbers are there between 30 and 40? (a) 4 (b) 1 (c) 2 (d) 3
31. 31.What is the smallest prime factor of 145? (a) 2 (b) 29 (c) 1 (d) 5
32. 32.What is the prime factorisation of 144? (a) 2⁵ × 3² (b) 2⁴ × 3² (c) 2³ × 3² (d) 2² × 3⁴
33. 33.How many factors does 58 have? (a) 3 (b) 4 (c) 2 (d) 5
34. 34.What is the prime factorisation of 80? (a) 2 × 5⁴ (b) 2⁵ × 5 (c) 2³ × 5 (d) 2⁴ × 5
35. 35.What is the prime factorisation of 28? (a) 2² × 7 (b) 2³ × 7 (c) 2 × 7² (d) 2 × 7
36. 36.How many factors does 86 have? (a) 2 (b) 3 (c) 4 (d) 5
37. 37.How many factors does 41 have? (a) 2 (b) 1 (c) 4 (d) 3
38. 38.How many factors does 98 have? (a) 7 (b) 4 (c) 6 (d) 5
39. 39.How many factors does 108 have? (a) 11 (b) 10 (c) 13 (d) 12
40. 40.What is the smallest prime factor of 169? (a) 3 (b) 1 (c) 13 (d) 2

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