Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 92 · 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 21 have? (a) 4 (b) 3 (c) 5 (d) 2
2. 2.How many prime numbers are there between 70 and 100? (a) 7 (b) 5 (c) 6 (d) 8
3. 3.What is the smallest prime factor of 65? (a) 1 (b) 2 (c) 5 (d) 13
4. 4.What is the smallest prime factor of 93? (a) 1 (b) 31 (c) 3 (d) 2
5. 5.How many prime numbers are there between 30 and 40? (a) 3 (b) 4 (c) 1 (d) 2
6. 6.What is the smallest prime factor of 143? (a) 11 (b) 2 (c) 13 (d) 1
7. 7.How many prime numbers are there between 10 and 40? (a) 9 (b) 7 (c) 8 (d) 10
8. 8.How many factors does 85 have? (a) 3 (b) 2 (c) 5 (d) 4
9. 9.How many prime numbers are there between 70 and 80? (a) 5 (b) 4 (c) 2 (d) 3
10. 10.What is the prime factorisation of 144? (a) 2³ × 3² (b) 2² × 3⁴ (c) 2⁵ × 3² (d) 2⁴ × 3²
11. 11.How many factors does 114 have? (a) 8 (b) 7 (c) 9 (d) 6
12. 12.How many prime numbers are there between 60 and 90? (a) 9 (b) 6 (c) 7 (d) 8
13. 13.How many prime numbers are there between 50 and 60? (a) 3 (b) 2 (c) 4 (d) 1
14. 14.What is the prime factorisation of 15? (a) 3 × 5 (b) 3³ × 5 (c) 3² × 5 (d) 3 × 5²
15. 15.How many factors does 98 have? (a) 7 (b) 6 (c) 4 (d) 5
16. 16.What is the smallest prime factor of 145? (a) 2 (b) 29 (c) 5 (d) 1
17. 17.What is the prime factorisation of 105? (a) 3² × 5 × 7 (b) 3 × 5 × 7 (c) 3³ × 5 × 7 (d) 3 × 5 × 7²
18. 18.What is the prime factorisation of 63? (a) 3 × 7² (b) 3³ × 7 (c) 3 × 7 (d) 3² × 7
19. 19.How many prime numbers are there between 80 and 100? (a) 3 (b) 2 (c) 4 (d) 5
20. 20.How many prime numbers are there between 70 and 90? (a) 4 (b) 6 (c) 5 (d) 7
21. 21.How many prime numbers are there between 80 and 110? (a) 6 (b) 9 (c) 8 (d) 7
22. 22.What is the prime factorisation of 18? (a) 2² × 3² (b) 2 × 3² (c) 2² × 3 (d) 2 × 3³
23. 23.What is the smallest prime factor of 87? (a) 3 (b) 1 (c) 29 (d) 2
24. 24.What is the prime factorisation of 140? (a) 2 × 5 × 7² (b) 2² × 5 × 7 (c) 2 × 5 × 7 (d) 2³ × 5 × 7
25. 25.How many factors does 25 have? (a) 1 (b) 3 (c) 2 (d) 4
26. 26.How many factors does 104 have? (a) 7 (b) 9 (c) 6 (d) 8
27. 27.What is the prime factorisation of 112? (a) 2⁴ × 7 (b) 2⁵ × 7 (c) 2³ × 7 (d) 2 × 7⁴
28. 28.What is the smallest prime factor of 77? (a) 2 (b) 7 (c) 11 (d) 1
29. 29.What is the smallest prime factor of 217? (a) 7 (b) 31 (c) 1 (d) 2
30. 30.How many prime numbers are there between 60 and 70? (a) 4 (b) 1 (c) 2 (d) 3
31. 31.How many prime numbers are there between 10 and 20? (a) 3 (b) 6 (c) 5 (d) 4
32. 32.What is the prime factorisation of 14? (a) 2² × 7 (b) 2³ × 7 (c) 2 × 7 (d) 2 × 7²
33. 33.How many factors does 26 have? (a) 3 (b) 2 (c) 4 (d) 5
34. 34.What is the prime factorisation of 80? (a) 2⁴ × 5 (b) 2³ × 5 (c) 2⁵ × 5 (d) 2 × 5⁴
35. 35.What is the smallest prime factor of 221? (a) 17 (b) 1 (c) 13 (d) 2
36. 36.What is the prime factorisation of 36? (a) 2 × 3² (b) 2³ × 3² (c) 2² × 3³ (d) 2² × 3²
37. 37.What is the prime factorisation of 315? (a) 3 × 5 × 7 (b) 3³ × 5 × 7 (c) 3 × 5 × 7² (d) 3² × 5 × 7
38. 38.What is the prime factorisation of 168? (a) 2⁴ × 3 × 7 (b) 2³ × 3 × 7 (c) 2 × 3 × 7³ (d) 2² × 3 × 7
39. 39.What is the smallest prime factor of 155? (a) 5 (b) 2 (c) 1 (d) 31
40. 40.What is the smallest prime factor of 403? (a) 13 (b) 1 (c) 31 (d) 2

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