Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 38 · 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 70 and 100? (a) 8 (b) 5 (c) 7 (d) 6
2. 2.How many factors does 79 have? (a) 3 (b) 4 (c) 1 (d) 2
3. 3.How many prime numbers are there between 90 and 100? (a) 3 (b) 2 (c) 1 (d) 0
4. 4.What is the prime factorisation of 210? (a) 2 × 3 × 5 × 7 (b) 2 × 3 × 5 × 7² (c) 2² × 3 × 5 × 7 (d) 2³ × 3 × 5 × 7
5. 5.How many prime numbers are there between 10 and 20? (a) 5 (b) 3 (c) 4 (d) 6
6. 6.What is the smallest prime factor of 69? (a) 1 (b) 23 (c) 2 (d) 3
7. 7.What is the smallest prime factor of 55? (a) 5 (b) 11 (c) 2 (d) 1
8. 8.How many factors does 32 have? (a) 5 (b) 7 (c) 6 (d) 4
9. 9.What is the smallest prime factor of 203? (a) 7 (b) 29 (c) 1 (d) 2
10. 10.How many prime numbers are there between 50 and 80? (a) 8 (b) 6 (c) 7 (d) 9
11. 11.What is the prime factorisation of 168? (a) 2 × 3 × 7³ (b) 2² × 3 × 7 (c) 2³ × 3 × 7 (d) 2⁴ × 3 × 7
12. 12.What is the prime factorisation of 12? (a) 2³ × 3 (b) 2² × 3 (c) 2 × 3 (d) 2 × 3²
13. 13.How many factors does 15 have? (a) 3 (b) 5 (c) 4 (d) 2
14. 14.How many factors does 65 have? (a) 4 (b) 5 (c) 3 (d) 2
15. 15.How many prime numbers are there between 40 and 50? (a) 5 (b) 2 (c) 3 (d) 4
16. 16.What is the smallest prime factor of 51? (a) 3 (b) 1 (c) 17 (d) 2
17. 17.How many factors does 60 have? (a) 11 (b) 10 (c) 12 (d) 13
18. 18.What is the prime factorisation of 45? (a) 3 × 5² (b) 3 × 5 (c) 3² × 5 (d) 3³ × 5
19. 19.How many prime numbers are there between 20 and 50? (a) 6 (b) 8 (c) 9 (d) 7
20. 20.How many factors does 47 have? (a) 3 (b) 4 (c) 2 (d) 1
21. 21.What is the prime factorisation of 80? (a) 2⁴ × 5 (b) 2³ × 5 (c) 2⁵ × 5 (d) 2 × 5⁴
22. 22.What is the smallest prime factor of 299? (a) 13 (b) 1 (c) 2 (d) 23
23. 23.What is the prime factorisation of 315? (a) 3³ × 5 × 7 (b) 3 × 5 × 7 (c) 3² × 5 × 7 (d) 3 × 5 × 7²
24. 24.What is the smallest prime factor of 87? (a) 29 (b) 1 (c) 2 (d) 3
25. 25.How many factors does 17 have? (a) 4 (b) 1 (c) 3 (d) 2
26. 26.What is the smallest prime factor of 221? (a) 1 (b) 17 (c) 2 (d) 13
27. 27.What is the smallest prime factor of 35? (a) 7 (b) 2 (c) 5 (d) 1
28. 28.What is the smallest prime factor of 77? (a) 11 (b) 2 (c) 7 (d) 1
29. 29.What is the prime factorisation of 30? (a) 2² × 3 × 5 (b) 2 × 3 × 5 (c) 2 × 3 × 5² (d) 2³ × 3 × 5
30. 30.How many factors does 112 have? (a) 9 (b) 8 (c) 10 (d) 11
31. 31.What is the prime factorisation of 84? (a) 2³ × 3 × 7 (b) 2 × 3 × 7 (c) 2² × 3 × 7 (d) 2 × 3 × 7²
32. 32.What is the prime factorisation of 120? (a) 2² × 3 × 5 (b) 2⁴ × 3 × 5 (c) 2 × 3 × 5³ (d) 2³ × 3 × 5
33. 33.How many factors does 31 have? (a) 4 (b) 2 (c) 1 (d) 3
34. 34.What is the prime factorisation of 240? (a) 2³ × 3 × 5 (b) 2 × 3 × 5⁴ (c) 2⁴ × 3 × 5 (d) 2⁵ × 3 × 5
35. 35.What is the prime factorisation of 40? (a) 2 × 5³ (b) 2⁴ × 5 (c) 2² × 5 (d) 2³ × 5
36. 36.How many factors does 41 have? (a) 1 (b) 2 (c) 4 (d) 3
37. 37.What is the prime factorisation of 56? (a) 2 × 7³ (b) 2² × 7 (c) 2⁴ × 7 (d) 2³ × 7
38. 38.What is the smallest prime factor of 143? (a) 13 (b) 1 (c) 11 (d) 2
39. 39.How many factors does 42 have? (a) 6 (b) 8 (c) 9 (d) 7
40. 40.How many prime numbers are there between 50 and 70? (a) 6 (b) 3 (c) 4 (d) 5

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