Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 85 · 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 60 and 90? (a) 6 (b) 9 (c) 7 (d) 8
2. 2.How many factors does 120 have? (a) 17 (b) 14 (c) 16 (d) 15
3. 3.How many prime numbers are there between 90 and 120? (a) 7 (b) 5 (c) 6 (d) 8
4. 4.What is the prime factorisation of 240? (a) 2⁴ × 3 × 5 (b) 2 × 3 × 5⁴ (c) 2⁵ × 3 × 5 (d) 2³ × 3 × 5
5. 5.What is the prime factorisation of 36? (a) 2 × 3² (b) 2² × 3³ (c) 2² × 3² (d) 2³ × 3²
6. 6.How many prime numbers are there between 10 and 40? (a) 9 (b) 7 (c) 8 (d) 10
7. 7.What is the prime factorisation of 84? (a) 2³ × 3 × 7 (b) 2 × 3 × 7 (c) 2 × 3 × 7² (d) 2² × 3 × 7
8. 8.What is the prime factorisation of 210? (a) 2² × 3 × 5 × 7 (b) 2 × 3 × 5 × 7² (c) 2³ × 3 × 5 × 7 (d) 2 × 3 × 5 × 7
9. 9.How many factors does 116 have? (a) 7 (b) 4 (c) 6 (d) 5
10. 10.How many factors does 83 have? (a) 4 (b) 1 (c) 3 (d) 2
11. 11.How many prime numbers are there between 10 and 30? (a) 8 (b) 6 (c) 5 (d) 7
12. 12.How many factors does 109 have? (a) 1 (b) 2 (c) 3 (d) 4
13. 13.What is the prime factorisation of 70? (a) 2³ × 5 × 7 (b) 2 × 5 × 7 (c) 2² × 5 × 7 (d) 2 × 5 × 7²
14. 14.What is the smallest prime factor of 133? (a) 1 (b) 7 (c) 19 (d) 2
15. 15.What is the smallest prime factor of 253? (a) 23 (b) 2 (c) 11 (d) 1
16. 16.How many factors does 98 have? (a) 7 (b) 4 (c) 6 (d) 5
17. 17.How many factors does 94 have? (a) 2 (b) 5 (c) 4 (d) 3
18. 18.How many prime numbers are there between 90 and 100? (a) 0 (b) 1 (c) 3 (d) 2
19. 19.How many prime numbers are there between 70 and 90? (a) 4 (b) 6 (c) 7 (d) 5
20. 20.What is the prime factorisation of 30? (a) 2 × 3 × 5² (b) 2 × 3 × 5 (c) 2³ × 3 × 5 (d) 2² × 3 × 5
21. 21.What is the prime factorisation of 35? (a) 5 × 7 (b) 5 × 7² (c) 5³ × 7 (d) 5² × 7
22. 22.What is the smallest prime factor of 49? (a) 2 (b) 1 (c) 7 (d) 3
23. 23.How many factors does 24 have? (a) 8 (b) 6 (c) 9 (d) 7
24. 24.How many factors does 76 have? (a) 6 (b) 4 (c) 7 (d) 5
25. 25.How many prime numbers are there between 20 and 30? (a) 3 (b) 2 (c) 1 (d) 4
26. 26.What is the prime factorisation of 48? (a) 2⁴ × 3 (b) 2⁵ × 3 (c) 2 × 3⁴ (d) 2³ × 3
27. 27.What is the prime factorisation of 504? (a) 2³ × 3² × 7 (b) 2 × 3² × 7³ (c) 2⁴ × 3² × 7 (d) 2² × 3² × 7
28. 28.How many factors does 42 have? (a) 7 (b) 9 (c) 8 (d) 6
29. 29.What is the prime factorisation of 144? (a) 2⁴ × 3² (b) 2⁵ × 3² (c) 2² × 3⁴ (d) 2³ × 3²
30. 30.What is the prime factorisation of 21? (a) 3³ × 7 (b) 3 × 7² (c) 3² × 7 (d) 3 × 7
31. 31.How many factors does 51 have? (a) 3 (b) 5 (c) 2 (d) 4
32. 32.How many prime numbers are there between 20 and 40? (a) 3 (b) 5 (c) 4 (d) 6
33. 33.What is the prime factorisation of 140? (a) 2 × 5 × 7 (b) 2³ × 5 × 7 (c) 2 × 5 × 7² (d) 2² × 5 × 7
34. 34.What is the prime factorisation of 126? (a) 2 × 3² × 7 (b) 2³ × 3² × 7 (c) 2 × 3² × 7² (d) 2² × 3² × 7
35. 35.What is the smallest prime factor of 77? (a) 2 (b) 1 (c) 7 (d) 11
36. 36.How many prime numbers are there between 30 and 40? (a) 1 (b) 4 (c) 2 (d) 3
37. 37.How many prime numbers are there between 60 and 70? (a) 2 (b) 3 (c) 4 (d) 1
38. 38.What is the smallest prime factor of 39? (a) 1 (b) 13 (c) 2 (d) 3
39. 39.What is the prime factorisation of 80? (a) 2⁵ × 5 (b) 2 × 5⁴ (c) 2⁴ × 5 (d) 2³ × 5
40. 40.What is the smallest prime factor of 145? (a) 5 (b) 1 (c) 2 (d) 29

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