Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 29 · 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 33 have? (a) 3 (b) 5 (c) 2 (d) 4
2. 2.What is the smallest prime factor of 187? (a) 17 (b) 1 (c) 2 (d) 11
3. 3.How many prime numbers are there between 90 and 110? (a) 4 (b) 6 (c) 7 (d) 5
4. 4.What is the prime factorisation of 144? (a) 2³ × 3² (b) 2² × 3⁴ (c) 2⁴ × 3² (d) 2⁵ × 3²
5. 5.What is the prime factorisation of 18? (a) 2 × 3² (b) 2² × 3 (c) 2 × 3³ (d) 2² × 3²
6. 6.What is the smallest prime factor of 91? (a) 1 (b) 2 (c) 7 (d) 13
7. 7.What is the smallest prime factor of 217? (a) 1 (b) 2 (c) 31 (d) 7
8. 8.How many factors does 41 have? (a) 3 (b) 2 (c) 4 (d) 1
9. 9.What is the smallest prime factor of 55? (a) 5 (b) 2 (c) 1 (d) 11
10. 10.How many factors does 72 have? (a) 12 (b) 13 (c) 11 (d) 10
11. 11.How many factors does 29 have? (a) 3 (b) 4 (c) 2 (d) 1
12. 12.How many factors does 103 have? (a) 3 (b) 1 (c) 2 (d) 4
13. 13.What is the prime factorisation of 240? (a) 2⁴ × 3 × 5 (b) 2⁵ × 3 × 5 (c) 2³ × 3 × 5 (d) 2 × 3 × 5⁴
14. 14.What is the prime factorisation of 60? (a) 2² × 3 × 5 (b) 2 × 3 × 5² (c) 2³ × 3 × 5 (d) 2 × 3 × 5
15. 15.How many factors does 48 have? (a) 8 (b) 9 (c) 11 (d) 10
16. 16.How many prime numbers are there between 30 and 50? (a) 4 (b) 5 (c) 7 (d) 6
17. 17.What is the prime factorisation of 180? (a) 2 × 3² × 5 (b) 2² × 3² × 5 (c) 2³ × 3² × 5 (d) 2 × 3² × 5²
18. 18.What is the prime factorisation of 28? (a) 2 × 7 (b) 2 × 7² (c) 2² × 7 (d) 2³ × 7
19. 19.How many factors does 64 have? (a) 6 (b) 5 (c) 8 (d) 7
20. 20.How many prime numbers are there between 20 and 40? (a) 3 (b) 5 (c) 6 (d) 4
21. 21.What is the smallest prime factor of 145? (a) 1 (b) 2 (c) 29 (d) 5
22. 22.What is the prime factorisation of 72? (a) 2⁴ × 3² (b) 2² × 3³ (c) 2³ × 3² (d) 2² × 3²
23. 23.How many factors does 17 have? (a) 1 (b) 3 (c) 4 (d) 2
24. 24.What is the smallest prime factor of 247? (a) 1 (b) 13 (c) 2 (d) 19
25. 25.What is the smallest prime factor of 377? (a) 2 (b) 13 (c) 1 (d) 29
26. 26.How many factors does 115 have? (a) 3 (b) 4 (c) 5 (d) 2
27. 27.What is the smallest prime factor of 115? (a) 1 (b) 23 (c) 5 (d) 2
28. 28.How many prime numbers are there between 50 and 70? (a) 5 (b) 3 (c) 4 (d) 6
29. 29.What is the prime factorisation of 20? (a) 2² × 5 (b) 2 × 5 (c) 2 × 5² (d) 2³ × 5
30. 30.How many prime numbers are there between 40 and 50? (a) 2 (b) 3 (c) 5 (d) 4
31. 31.How many factors does 63 have? (a) 4 (b) 5 (c) 7 (d) 6
32. 32.How many prime numbers are there between 70 and 100? (a) 7 (b) 8 (c) 5 (d) 6
33. 33.How many prime numbers are there between 80 and 100? (a) 5 (b) 3 (c) 4 (d) 2
34. 34.What is the prime factorisation of 21? (a) 3³ × 7 (b) 3² × 7 (c) 3 × 7 (d) 3 × 7²
35. 35.What is the smallest prime factor of 87? (a) 2 (b) 29 (c) 3 (d) 1
36. 36.How many prime numbers are there between 60 and 70? (a) 4 (b) 2 (c) 3 (d) 1
37. 37.What is the prime factorisation of 24? (a) 2 × 3³ (b) 2² × 3 (c) 2³ × 3 (d) 2⁴ × 3
38. 38.How many prime numbers are there between 70 and 80? (a) 2 (b) 5 (c) 4 (d) 3
39. 39.How many prime numbers are there between 10 and 40? (a) 8 (b) 10 (c) 7 (d) 9
40. 40.What is the smallest prime factor of 51? (a) 2 (b) 3 (c) 1 (d) 17

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