Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 91 · 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 factors does 78 have? (a) 6 (b) 9 (c) 8 (d) 7
2. 2.What is the prime factorisation of 504? (a) 2⁴ × 3² × 7 (b) 2² × 3² × 7 (c) 2³ × 3² × 7 (d) 2 × 3² × 7³
3. 3.How many prime numbers are there between 40 and 50? (a) 2 (b) 3 (c) 5 (d) 4
4. 4.How many prime numbers are there between 10 and 30? (a) 7 (b) 8 (c) 6 (d) 5
5. 5.How many factors does 93 have? (a) 2 (b) 4 (c) 5 (d) 3
6. 6.What is the smallest prime factor of 87? (a) 3 (b) 2 (c) 29 (d) 1
7. 7.How many factors does 73 have? (a) 1 (b) 2 (c) 4 (d) 3
8. 8.How many prime numbers are there between 20 and 30? (a) 4 (b) 3 (c) 2 (d) 1
9. 9.What is the prime factorisation of 30? (a) 2 × 3 × 5² (b) 2² × 3 × 5 (c) 2 × 3 × 5 (d) 2³ × 3 × 5
10. 10.How many factors does 100 have? (a) 9 (b) 10 (c) 7 (d) 8
11. 11.What is the smallest prime factor of 35? (a) 7 (b) 1 (c) 2 (d) 5
12. 12.How many prime numbers are there between 30 and 60? (a) 7 (b) 9 (c) 6 (d) 8
13. 13.What is the prime factorisation of 35? (a) 5³ × 7 (b) 5 × 7² (c) 5 × 7 (d) 5² × 7
14. 14.What is the smallest prime factor of 55? (a) 11 (b) 5 (c) 2 (d) 1
15. 15.How many prime numbers are there between 50 and 80? (a) 9 (b) 7 (c) 8 (d) 6
16. 16.How many factors does 117 have? (a) 7 (b) 5 (c) 4 (d) 6
17. 17.How many prime numbers are there between 80 and 100? (a) 3 (b) 5 (c) 4 (d) 2
18. 18.How many prime numbers are there between 90 and 100? (a) 0 (b) 1 (c) 3 (d) 2
19. 19.What is the prime factorisation of 45? (a) 3² × 5 (b) 3 × 5² (c) 3³ × 5 (d) 3 × 5
20. 20.What is the prime factorisation of 63? (a) 3 × 7² (b) 3 × 7 (c) 3³ × 7 (d) 3² × 7
21. 21.What is the prime factorisation of 168? (a) 2² × 3 × 7 (b) 2⁴ × 3 × 7 (c) 2 × 3 × 7³ (d) 2³ × 3 × 7
22. 22.How many factors does 76 have? (a) 5 (b) 4 (c) 6 (d) 7
23. 23.How many factors does 60 have? (a) 11 (b) 13 (c) 12 (d) 10
24. 24.How many prime numbers are there between 80 and 110? (a) 9 (b) 7 (c) 6 (d) 8
25. 25.What is the smallest prime factor of 95? (a) 2 (b) 19 (c) 5 (d) 1
26. 26.How many prime numbers are there between 50 and 60? (a) 4 (b) 1 (c) 3 (d) 2
27. 27.What is the prime factorisation of 36? (a) 2 × 3² (b) 2² × 3³ (c) 2³ × 3² (d) 2² × 3²
28. 28.How many prime numbers are there between 30 and 40? (a) 4 (b) 2 (c) 1 (d) 3
29. 29.How many factors does 35 have? (a) 4 (b) 2 (c) 5 (d) 3
30. 30.What is the smallest prime factor of 403? (a) 31 (b) 13 (c) 1 (d) 2
31. 31.How many factors does 63 have? (a) 5 (b) 6 (c) 7 (d) 4
32. 32.What is the prime factorisation of 112? (a) 2⁵ × 7 (b) 2 × 7⁴ (c) 2⁴ × 7 (d) 2³ × 7
33. 33.What is the prime factorisation of 120? (a) 2⁴ × 3 × 5 (b) 2² × 3 × 5 (c) 2 × 3 × 5³ (d) 2³ × 3 × 5
34. 34.How many factors does 56 have? (a) 6 (b) 8 (c) 9 (d) 7
35. 35.What is the prime factorisation of 420? (a) 2 × 3 × 5 × 7 (b) 2² × 3 × 5 × 7 (c) 2 × 3 × 5 × 7² (d) 2³ × 3 × 5 × 7
36. 36.What is the smallest prime factor of 91? (a) 2 (b) 1 (c) 13 (d) 7
37. 37.What is the smallest prime factor of 69? (a) 1 (b) 23 (c) 3 (d) 2
38. 38.How many factors does 55 have? (a) 3 (b) 2 (c) 4 (d) 5
39. 39.How many factors does 97 have? (a) 2 (b) 3 (c) 4 (d) 1
40. 40.How many factors does 37 have? (a) 1 (b) 3 (c) 4 (d) 2

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