Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 197 · 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 50 and 80? (a) 7 (b) 8 (c) 9 (d) 6
2. 2.How many factors does 17 have? (a) 2 (b) 1 (c) 4 (d) 3
3. 3.What is the prime factorisation of 126? (a) 2 × 3² × 7 (b) 2 × 3² × 7² (c) 2³ × 3² × 7 (d) 2² × 3² × 7
4. 4.What is the smallest prime factor of 69? (a) 1 (b) 23 (c) 3 (d) 2
5. 5.How many factors does 83 have? (a) 1 (b) 4 (c) 3 (d) 2
6. 6.How many factors does 104 have? (a) 9 (b) 7 (c) 6 (d) 8
7. 7.How many prime numbers are there between 10 and 40? (a) 10 (b) 8 (c) 9 (d) 7
8. 8.What is the prime factorisation of 84? (a) 2² × 3 × 7 (b) 2³ × 3 × 7 (c) 2 × 3 × 7² (d) 2 × 3 × 7
9. 9.How many prime numbers are there between 80 and 90? (a) 1 (b) 2 (c) 3 (d) 4
10. 10.What is the smallest prime factor of 217? (a) 31 (b) 1 (c) 7 (d) 2
11. 11.What is the smallest prime factor of 49? (a) 2 (b) 3 (c) 1 (d) 7
12. 12.How many factors does 16 have? (a) 6 (b) 3 (c) 4 (d) 5
13. 13.How many factors does 24 have? (a) 9 (b) 7 (c) 6 (d) 8
14. 14.What is the prime factorisation of 20? (a) 2 × 5 (b) 2 × 5² (c) 2² × 5 (d) 2³ × 5
15. 15.How many factors does 109 have? (a) 1 (b) 2 (c) 3 (d) 4
16. 16.What is the smallest prime factor of 91? (a) 1 (b) 2 (c) 7 (d) 13
17. 17.How many factors does 88 have? (a) 6 (b) 7 (c) 9 (d) 8
18. 18.What is the prime factorisation of 315? (a) 3³ × 5 × 7 (b) 3 × 5 × 7² (c) 3 × 5 × 7 (d) 3² × 5 × 7
19. 19.How many factors does 116 have? (a) 5 (b) 6 (c) 4 (d) 7
20. 20.How many factors does 91 have? (a) 4 (b) 5 (c) 2 (d) 3
21. 21.What is the prime factorisation of 24? (a) 2 × 3³ (b) 2⁴ × 3 (c) 2³ × 3 (d) 2² × 3
22. 22.What is the prime factorisation of 35? (a) 5 × 7 (b) 5 × 7² (c) 5³ × 7 (d) 5² × 7
23. 23.What is the prime factorisation of 140? (a) 2 × 5 × 7 (b) 2³ × 5 × 7 (c) 2² × 5 × 7 (d) 2 × 5 × 7²
24. 24.How many factors does 77 have? (a) 2 (b) 3 (c) 4 (d) 5
25. 25.How many prime numbers are there between 80 and 110? (a) 7 (b) 6 (c) 8 (d) 9
26. 26.How many factors does 57 have? (a) 4 (b) 2 (c) 5 (d) 3
27. 27.How many prime numbers are there between 60 and 90? (a) 6 (b) 8 (c) 7 (d) 9
28. 28.What is the prime factorisation of 28? (a) 2 × 7² (b) 2³ × 7 (c) 2 × 7 (d) 2² × 7
29. 29.How many prime numbers are there between 20 and 30? (a) 4 (b) 3 (c) 1 (d) 2
30. 30.What is the prime factorisation of 72? (a) 2⁴ × 3² (b) 2² × 3³ (c) 2³ × 3² (d) 2² × 3²
31. 31.What is the prime factorisation of 30? (a) 2 × 3 × 5 (b) 2² × 3 × 5 (c) 2³ × 3 × 5 (d) 2 × 3 × 5²
32. 32.How many factors does 14 have? (a) 4 (b) 5 (c) 3 (d) 2
33. 33.What is the prime factorisation of 90? (a) 2³ × 3² × 5 (b) 2² × 3² × 5 (c) 2 × 3² × 5 (d) 2 × 3² × 5²
34. 34.How many factors does 12 have? (a) 6 (b) 4 (c) 5 (d) 7
35. 35.What is the smallest prime factor of 57? (a) 19 (b) 1 (c) 3 (d) 2
36. 36.How many factors does 95 have? (a) 2 (b) 3 (c) 5 (d) 4
37. 37.How many factors does 60 have? (a) 12 (b) 13 (c) 10 (d) 11
38. 38.How many factors does 31 have? (a) 1 (b) 4 (c) 2 (d) 3
39. 39.How many prime numbers are there between 90 and 110? (a) 6 (b) 4 (c) 7 (d) 5
40. 40.How many prime numbers are there between 40 and 70? (a) 8 (b) 6 (c) 7 (d) 9

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