Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 125 · 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 smallest prime factor of 77? (a) 1 (b) 7 (c) 11 (d) 2
2. 2.What is the prime factorisation of 24? (a) 2² × 3 (b) 2⁴ × 3 (c) 2 × 3³ (d) 2³ × 3
3. 3.What is the prime factorisation of 168? (a) 2³ × 3 × 7 (b) 2⁴ × 3 × 7 (c) 2² × 3 × 7 (d) 2 × 3 × 7³
4. 4.How many prime numbers are there between 30 and 40? (a) 3 (b) 4 (c) 1 (d) 2
5. 5.How many prime numbers are there between 80 and 90? (a) 1 (b) 3 (c) 4 (d) 2
6. 6.How many factors does 19 have? (a) 4 (b) 3 (c) 1 (d) 2
7. 7.What is the smallest prime factor of 217? (a) 2 (b) 1 (c) 7 (d) 31
8. 8.How many factors does 99 have? (a) 7 (b) 4 (c) 5 (d) 6
9. 9.What is the prime factorisation of 105? (a) 3 × 5 × 7² (b) 3² × 5 × 7 (c) 3³ × 5 × 7 (d) 3 × 5 × 7
10. 10.What is the prime factorisation of 315? (a) 3 × 5 × 7 (b) 3 × 5 × 7² (c) 3³ × 5 × 7 (d) 3² × 5 × 7
11. 11.What is the smallest prime factor of 65? (a) 13 (b) 2 (c) 5 (d) 1
12. 12.How many prime numbers are there between 50 and 80? (a) 9 (b) 8 (c) 7 (d) 6
13. 13.What is the smallest prime factor of 57? (a) 19 (b) 2 (c) 3 (d) 1
14. 14.How many prime numbers are there between 30 and 60? (a) 9 (b) 7 (c) 6 (d) 8
15. 15.How many prime numbers are there between 20 and 40? (a) 4 (b) 5 (c) 6 (d) 3
16. 16.What is the smallest prime factor of 299? (a) 2 (b) 13 (c) 23 (d) 1
17. 17.How many factors does 85 have? (a) 3 (b) 4 (c) 5 (d) 2
18. 18.What is the smallest prime factor of 319? (a) 1 (b) 2 (c) 11 (d) 29
19. 19.How many factors does 84 have? (a) 11 (b) 13 (c) 12 (d) 10
20. 20.What is the prime factorisation of 240? (a) 2⁴ × 3 × 5 (b) 2 × 3 × 5⁴ (c) 2⁵ × 3 × 5 (d) 2³ × 3 × 5
21. 21.How many factors does 93 have? (a) 5 (b) 2 (c) 3 (d) 4
22. 22.What is the smallest prime factor of 21? (a) 2 (b) 3 (c) 7 (d) 1
23. 23.How many factors does 31 have? (a) 2 (b) 4 (c) 3 (d) 1
24. 24.How many prime numbers are there between 50 and 70? (a) 6 (b) 5 (c) 4 (d) 3
25. 25.How many prime numbers are there between 70 and 80? (a) 4 (b) 5 (c) 2 (d) 3
26. 26.What is the prime factorisation of 45? (a) 3³ × 5 (b) 3 × 5 (c) 3 × 5² (d) 3² × 5
27. 27.How many prime numbers are there between 60 and 90? (a) 6 (b) 9 (c) 7 (d) 8
28. 28.What is the smallest prime factor of 51? (a) 3 (b) 17 (c) 2 (d) 1
29. 29.What is the smallest prime factor of 133? (a) 7 (b) 19 (c) 2 (d) 1
30. 30.What is the prime factorisation of 18? (a) 2² × 3² (b) 2 × 3³ (c) 2² × 3 (d) 2 × 3²
31. 31.How many factors does 112 have? (a) 9 (b) 10 (c) 8 (d) 11
32. 32.How many factors does 109 have? (a) 1 (b) 3 (c) 4 (d) 2
33. 33.What is the prime factorisation of 112? (a) 2⁵ × 7 (b) 2³ × 7 (c) 2 × 7⁴ (d) 2⁴ × 7
34. 34.How many factors does 37 have? (a) 1 (b) 4 (c) 3 (d) 2
35. 35.What is the smallest prime factor of 55? (a) 1 (b) 2 (c) 11 (d) 5
36. 36.What is the prime factorisation of 40? (a) 2⁴ × 5 (b) 2 × 5³ (c) 2³ × 5 (d) 2² × 5
37. 37.What is the smallest prime factor of 253? (a) 1 (b) 23 (c) 11 (d) 2
38. 38.How many prime numbers are there between 20 and 50? (a) 8 (b) 7 (c) 6 (d) 9
39. 39.How many factors does 107 have? (a) 4 (b) 3 (c) 1 (d) 2
40. 40.How many factors does 39 have? (a) 2 (b) 3 (c) 4 (d) 5

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