Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 43 · 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 prime factorisation of 20? (a) 2 × 5 (b) 2³ × 5 (c) 2² × 5 (d) 2 × 5²
2. 2.How many factors does 40 have? (a) 9 (b) 6 (c) 7 (d) 8
3. 3.How many factors does 93 have? (a) 3 (b) 2 (c) 4 (d) 5
4. 4.What is the prime factorisation of 144? (a) 2³ × 3² (b) 2⁴ × 3² (c) 2² × 3⁴ (d) 2⁵ × 3²
5. 5.What is the prime factorisation of 35? (a) 5³ × 7 (b) 5 × 7² (c) 5 × 7 (d) 5² × 7
6. 6.How many factors does 78 have? (a) 8 (b) 9 (c) 6 (d) 7
7. 7.How many prime numbers are there between 90 and 120? (a) 7 (b) 8 (c) 6 (d) 5
8. 8.What is the prime factorisation of 240? (a) 2 × 3 × 5⁴ (b) 2³ × 3 × 5 (c) 2⁴ × 3 × 5 (d) 2⁵ × 3 × 5
9. 9.What is the smallest prime factor of 187? (a) 2 (b) 1 (c) 11 (d) 17
10. 10.How many prime numbers are there between 70 and 80? (a) 4 (b) 5 (c) 3 (d) 2
11. 11.What is the prime factorisation of 126? (a) 2² × 3² × 7 (b) 2³ × 3² × 7 (c) 2 × 3² × 7 (d) 2 × 3² × 7²
12. 12.What is the prime factorisation of 56? (a) 2 × 7³ (b) 2⁴ × 7 (c) 2³ × 7 (d) 2² × 7
13. 13.What is the prime factorisation of 18? (a) 2² × 3 (b) 2 × 3³ (c) 2² × 3² (d) 2 × 3²
14. 14.What is the smallest prime factor of 161? (a) 1 (b) 7 (c) 23 (d) 2
15. 15.How many prime numbers are there between 80 and 110? (a) 6 (b) 9 (c) 8 (d) 7
16. 16.What is the smallest prime factor of 221? (a) 17 (b) 2 (c) 1 (d) 13
17. 17.How many factors does 103 have? (a) 2 (b) 3 (c) 1 (d) 4
18. 18.How many prime numbers are there between 30 and 40? (a) 2 (b) 3 (c) 1 (d) 4
19. 19.What is the smallest prime factor of 33? (a) 11 (b) 2 (c) 3 (d) 1
20. 20.What is the prime factorisation of 360? (a) 2⁴ × 3² × 5 (b) 2 × 3² × 5³ (c) 2² × 3² × 5 (d) 2³ × 3² × 5
21. 21.What is the prime factorisation of 210? (a) 2 × 3 × 5 × 7² (b) 2 × 3 × 5 × 7 (c) 2³ × 3 × 5 × 7 (d) 2² × 3 × 5 × 7
22. 22.How many factors does 106 have? (a) 3 (b) 2 (c) 5 (d) 4
23. 23.What is the prime factorisation of 14? (a) 2³ × 7 (b) 2 × 7 (c) 2 × 7² (d) 2² × 7
24. 24.How many factors does 97 have? (a) 1 (b) 3 (c) 2 (d) 4
25. 25.How many factors does 84 have? (a) 13 (b) 11 (c) 10 (d) 12
26. 26.How many factors does 111 have? (a) 3 (b) 5 (c) 4 (d) 2
27. 27.What is the smallest prime factor of 95? (a) 19 (b) 2 (c) 5 (d) 1
28. 28.What is the smallest prime factor of 319? (a) 1 (b) 29 (c) 2 (d) 11
29. 29.What is the smallest prime factor of 217? (a) 2 (b) 1 (c) 7 (d) 31
30. 30.How many factors does 19 have? (a) 1 (b) 2 (c) 4 (d) 3
31. 31.What is the smallest prime factor of 133? (a) 2 (b) 1 (c) 19 (d) 7
32. 32.What is the smallest prime factor of 93? (a) 31 (b) 2 (c) 3 (d) 1
33. 33.How many prime numbers are there between 90 and 100? (a) 0 (b) 1 (c) 3 (d) 2
34. 34.How many prime numbers are there between 60 and 90? (a) 6 (b) 8 (c) 9 (d) 7
35. 35.How many factors does 74 have? (a) 5 (b) 2 (c) 4 (d) 3
36. 36.What is the prime factorisation of 105? (a) 3 × 5 × 7² (b) 3 × 5 × 7 (c) 3² × 5 × 7 (d) 3³ × 5 × 7
37. 37.What is the prime factorisation of 28? (a) 2³ × 7 (b) 2 × 7² (c) 2² × 7 (d) 2 × 7
38. 38.How many factors does 89 have? (a) 4 (b) 1 (c) 3 (d) 2
39. 39.How many prime numbers are there between 80 and 100? (a) 4 (b) 5 (c) 3 (d) 2
40. 40.What is the prime factorisation of 120? (a) 2 × 3 × 5³ (b) 2² × 3 × 5 (c) 2⁴ × 3 × 5 (d) 2³ × 3 × 5

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