Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 148 · 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 35? (a) 2 (b) 1 (c) 7 (d) 5
2. 2.What is the prime factorisation of 48? (a) 2⁴ × 3 (b) 2³ × 3 (c) 2 × 3⁴ (d) 2⁵ × 3
3. 3.What is the smallest prime factor of 21? (a) 1 (b) 3 (c) 7 (d) 2
4. 4.What is the prime factorisation of 20? (a) 2 × 5² (b) 2 × 5 (c) 2² × 5 (d) 2³ × 5
5. 5.How many factors does 91 have? (a) 4 (b) 5 (c) 3 (d) 2
6. 6.What is the prime factorisation of 21? (a) 3³ × 7 (b) 3 × 7² (c) 3² × 7 (d) 3 × 7
7. 7.What is the smallest prime factor of 247? (a) 2 (b) 13 (c) 1 (d) 19
8. 8.What is the prime factorisation of 210? (a) 2³ × 3 × 5 × 7 (b) 2 × 3 × 5 × 7 (c) 2 × 3 × 5 × 7² (d) 2² × 3 × 5 × 7
9. 9.How many factors does 13 have? (a) 3 (b) 1 (c) 4 (d) 2
10. 10.How many factors does 31 have? (a) 1 (b) 4 (c) 3 (d) 2
11. 11.What is the prime factorisation of 280? (a) 2⁴ × 5 × 7 (b) 2 × 5 × 7³ (c) 2³ × 5 × 7 (d) 2² × 5 × 7
12. 12.How many prime numbers are there between 30 and 60? (a) 6 (b) 9 (c) 8 (d) 7
13. 13.How many factors does 63 have? (a) 7 (b) 6 (c) 4 (d) 5
14. 14.What is the prime factorisation of 168? (a) 2 × 3 × 7³ (b) 2⁴ × 3 × 7 (c) 2³ × 3 × 7 (d) 2² × 3 × 7
15. 15.How many factors does 44 have? (a) 6 (b) 4 (c) 5 (d) 7
16. 16.What is the prime factorisation of 36? (a) 2 × 3² (b) 2² × 3³ (c) 2² × 3² (d) 2³ × 3²
17. 17.How many prime numbers are there between 20 and 30? (a) 2 (b) 3 (c) 1 (d) 4
18. 18.How many prime numbers are there between 90 and 100? (a) 2 (b) 3 (c) 0 (d) 1
19. 19.What is the smallest prime factor of 33? (a) 1 (b) 2 (c) 11 (d) 3
20. 20.How many prime numbers are there between 80 and 90? (a) 4 (b) 2 (c) 1 (d) 3
21. 21.How many prime numbers are there between 20 and 40? (a) 6 (b) 5 (c) 4 (d) 3
22. 22.What is the prime factorisation of 112? (a) 2 × 7⁴ (b) 2⁴ × 7 (c) 2³ × 7 (d) 2⁵ × 7
23. 23.What is the prime factorisation of 144? (a) 2² × 3⁴ (b) 2³ × 3² (c) 2⁵ × 3² (d) 2⁴ × 3²
24. 24.What is the prime factorisation of 420? (a) 2² × 3 × 5 × 7 (b) 2 × 3 × 5 × 7 (c) 2 × 3 × 5 × 7² (d) 2³ × 3 × 5 × 7
25. 25.How many factors does 95 have? (a) 2 (b) 5 (c) 3 (d) 4
26. 26.How many factors does 105 have? (a) 8 (b) 6 (c) 9 (d) 7
27. 27.How many factors does 25 have? (a) 2 (b) 4 (c) 1 (d) 3
28. 28.What is the smallest prime factor of 91? (a) 2 (b) 13 (c) 1 (d) 7
29. 29.What is the prime factorisation of 60? (a) 2² × 3 × 5 (b) 2 × 3 × 5 (c) 2 × 3 × 5² (d) 2³ × 3 × 5
30. 30.How many factors does 81 have? (a) 6 (b) 4 (c) 5 (d) 3
31. 31.What is the smallest prime factor of 69? (a) 3 (b) 1 (c) 2 (d) 23
32. 32.How many prime numbers are there between 40 and 60? (a) 6 (b) 5 (c) 4 (d) 7
33. 33.What is the smallest prime factor of 95? (a) 19 (b) 1 (c) 5 (d) 2
34. 34.How many factors does 30 have? (a) 6 (b) 7 (c) 9 (d) 8
35. 35.How many factors does 73 have? (a) 4 (b) 2 (c) 3 (d) 1
36. 36.What is the prime factorisation of 12? (a) 2 × 3 (b) 2 × 3² (c) 2² × 3 (d) 2³ × 3
37. 37.What is the prime factorisation of 504? (a) 2 × 3² × 7³ (b) 2² × 3² × 7 (c) 2⁴ × 3² × 7 (d) 2³ × 3² × 7
38. 38.What is the prime factorisation of 70? (a) 2 × 5 × 7² (b) 2³ × 5 × 7 (c) 2² × 5 × 7 (d) 2 × 5 × 7
39. 39.What is the prime factorisation of 72? (a) 2² × 3³ (b) 2⁴ × 3² (c) 2³ × 3² (d) 2² × 3²
40. 40.What is the smallest prime factor of 187? (a) 17 (b) 1 (c) 2 (d) 11

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