Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 127 · 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 209? (a) 19 (b) 11 (c) 2 (d) 1
2. 2.What is the smallest prime factor of 145? (a) 2 (b) 5 (c) 29 (d) 1
3. 3.What is the smallest prime factor of 217? (a) 31 (b) 1 (c) 2 (d) 7
4. 4.What is the smallest prime factor of 49? (a) 2 (b) 3 (c) 7 (d) 1
5. 5.How many factors does 59 have? (a) 2 (b) 4 (c) 1 (d) 3
6. 6.What is the prime factorisation of 30? (a) 2 × 3 × 5 (b) 2 × 3 × 5² (c) 2² × 3 × 5 (d) 2³ × 3 × 5
7. 7.What is the prime factorisation of 36? (a) 2³ × 3² (b) 2² × 3² (c) 2² × 3³ (d) 2 × 3²
8. 8.How many factors does 66 have? (a) 6 (b) 8 (c) 9 (d) 7
9. 9.What is the prime factorisation of 70? (a) 2³ × 5 × 7 (b) 2 × 5 × 7² (c) 2 × 5 × 7 (d) 2² × 5 × 7
10. 10.What is the smallest prime factor of 69? (a) 23 (b) 1 (c) 3 (d) 2
11. 11.What is the smallest prime factor of 143? (a) 1 (b) 2 (c) 11 (d) 13
12. 12.How many factors does 101 have? (a) 4 (b) 3 (c) 1 (d) 2
13. 13.What is the prime factorisation of 63? (a) 3 × 7² (b) 3 × 7 (c) 3³ × 7 (d) 3² × 7
14. 14.How many prime numbers are there between 40 and 70? (a) 7 (b) 6 (c) 8 (d) 9
15. 15.How many factors does 108 have? (a) 11 (b) 12 (c) 13 (d) 10
16. 16.What is the smallest prime factor of 85? (a) 17 (b) 5 (c) 2 (d) 1
17. 17.What is the prime factorisation of 45? (a) 3 × 5² (b) 3² × 5 (c) 3 × 5 (d) 3³ × 5
18. 18.How many factors does 112 have? (a) 8 (b) 11 (c) 10 (d) 9
19. 19.How many prime numbers are there between 60 and 90? (a) 7 (b) 8 (c) 9 (d) 6
20. 20.How many prime numbers are there between 70 and 80? (a) 2 (b) 5 (c) 3 (d) 4
21. 21.How many factors does 113 have? (a) 4 (b) 3 (c) 2 (d) 1
22. 22.What is the prime factorisation of 40? (a) 2⁴ × 5 (b) 2³ × 5 (c) 2 × 5³ (d) 2² × 5
23. 23.What is the prime factorisation of 80? (a) 2 × 5⁴ (b) 2⁴ × 5 (c) 2⁵ × 5 (d) 2³ × 5
24. 24.What is the smallest prime factor of 95? (a) 19 (b) 2 (c) 5 (d) 1
25. 25.What is the prime factorisation of 60? (a) 2 × 3 × 5² (b) 2 × 3 × 5 (c) 2³ × 3 × 5 (d) 2² × 3 × 5
26. 26.How many prime numbers are there between 10 and 40? (a) 9 (b) 10 (c) 8 (d) 7
27. 27.What is the prime factorisation of 504? (a) 2² × 3² × 7 (b) 2 × 3² × 7³ (c) 2⁴ × 3² × 7 (d) 2³ × 3² × 7
28. 28.What is the smallest prime factor of 21? (a) 1 (b) 3 (c) 2 (d) 7
29. 29.What is the prime factorisation of 42? (a) 2² × 3 × 7 (b) 2 × 3 × 7 (c) 2³ × 3 × 7 (d) 2 × 3 × 7²
30. 30.How many prime numbers are there between 90 and 110? (a) 5 (b) 4 (c) 6 (d) 7
31. 31.What is the smallest prime factor of 119? (a) 2 (b) 7 (c) 1 (d) 17
32. 32.What is the smallest prime factor of 35? (a) 7 (b) 1 (c) 2 (d) 5
33. 33.How many factors does 14 have? (a) 3 (b) 5 (c) 2 (d) 4
34. 34.What is the smallest prime factor of 39? (a) 13 (b) 2 (c) 3 (d) 1
35. 35.What is the prime factorisation of 35? (a) 5² × 7 (b) 5 × 7 (c) 5 × 7² (d) 5³ × 7
36. 36.How many prime numbers are there between 20 and 30? (a) 2 (b) 3 (c) 4 (d) 1
37. 37.How many factors does 57 have? (a) 2 (b) 5 (c) 3 (d) 4
38. 38.How many prime numbers are there between 40 and 50? (a) 4 (b) 5 (c) 2 (d) 3
39. 39.How many prime numbers are there between 60 and 70? (a) 4 (b) 1 (c) 2 (d) 3
40. 40.What is the prime factorisation of 12? (a) 2² × 3 (b) 2 × 3 (c) 2 × 3² (d) 2³ × 3

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