Prime Numbers and Factors Worksheet for Olympiad

Maths Olympiad topic 31: Primes and divisors · Set 78 · 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 85? (a) 17 (b) 2 (c) 5 (d) 1
2. 2.How many factors does 70 have? (a) 6 (b) 9 (c) 7 (d) 8
3. 3.How many prime numbers are there between 60 and 70? (a) 4 (b) 1 (c) 2 (d) 3
4. 4.How many factors does 34 have? (a) 5 (b) 2 (c) 4 (d) 3
5. 5.How many prime numbers are there between 50 and 70? (a) 3 (b) 5 (c) 4 (d) 6
6. 6.How many prime numbers are there between 30 and 40? (a) 2 (b) 4 (c) 1 (d) 3
7. 7.What is the prime factorisation of 420? (a) 2³ × 3 × 5 × 7 (b) 2 × 3 × 5 × 7 (c) 2 × 3 × 5 × 7² (d) 2² × 3 × 5 × 7
8. 8.What is the smallest prime factor of 21? (a) 7 (b) 3 (c) 2 (d) 1
9. 9.How many factors does 112 have? (a) 8 (b) 10 (c) 9 (d) 11
10. 10.What is the smallest prime factor of 91? (a) 13 (b) 2 (c) 1 (d) 7
11. 11.How many prime numbers are there between 40 and 70? (a) 8 (b) 6 (c) 9 (d) 7
12. 12.What is the smallest prime factor of 55? (a) 2 (b) 11 (c) 5 (d) 1
13. 13.What is the smallest prime factor of 51? (a) 17 (b) 2 (c) 1 (d) 3
14. 14.How many factors does 105 have? (a) 7 (b) 9 (c) 8 (d) 6
15. 15.How many prime numbers are there between 70 and 100? (a) 8 (b) 7 (c) 5 (d) 6
16. 16.What is the smallest prime factor of 319? (a) 29 (b) 1 (c) 2 (d) 11
17. 17.What is the prime factorisation of 105? (a) 3² × 5 × 7 (b) 3 × 5 × 7² (c) 3 × 5 × 7 (d) 3³ × 5 × 7
18. 18.How many prime numbers are there between 80 and 110? (a) 9 (b) 6 (c) 7 (d) 8
19. 19.How many prime numbers are there between 80 and 90? (a) 1 (b) 2 (c) 4 (d) 3
20. 20.What is the smallest prime factor of 95? (a) 2 (b) 5 (c) 1 (d) 19
21. 21.How many factors does 47 have? (a) 1 (b) 3 (c) 2 (d) 4
22. 22.What is the smallest prime factor of 161? (a) 7 (b) 1 (c) 2 (d) 23
23. 23.What is the prime factorisation of 252? (a) 2 × 3² × 7² (b) 2 × 3² × 7 (c) 2² × 3² × 7 (d) 2³ × 3² × 7
24. 24.What is the prime factorisation of 180? (a) 2³ × 3² × 5 (b) 2² × 3² × 5 (c) 2 × 3² × 5 (d) 2 × 3² × 5²
25. 25.What is the smallest prime factor of 145? (a) 29 (b) 1 (c) 5 (d) 2
26. 26.How many prime numbers are there between 20 and 50? (a) 9 (b) 6 (c) 8 (d) 7
27. 27.What is the smallest prime factor of 35? (a) 1 (b) 2 (c) 7 (d) 5
28. 28.What is the smallest prime factor of 121? (a) 1 (b) 11 (c) 2 (d) 3
29. 29.How many factors does 19 have? (a) 3 (b) 4 (c) 1 (d) 2
30. 30.What is the prime factorisation of 336? (a) 2⁵ × 3 × 7 (b) 2 × 3 × 7⁴ (c) 2⁴ × 3 × 7 (d) 2³ × 3 × 7
31. 31.What is the prime factorisation of 126? (a) 2² × 3² × 7 (b) 2 × 3² × 7² (c) 2³ × 3² × 7 (d) 2 × 3² × 7
32. 32.What is the prime factorisation of 112? (a) 2 × 7⁴ (b) 2³ × 7 (c) 2⁵ × 7 (d) 2⁴ × 7
33. 33.How many prime numbers are there between 90 and 110? (a) 4 (b) 7 (c) 6 (d) 5
34. 34.How many prime numbers are there between 60 and 80? (a) 5 (b) 7 (c) 4 (d) 6
35. 35.How many prime numbers are there between 10 and 20? (a) 4 (b) 6 (c) 3 (d) 5
36. 36.How many prime numbers are there between 20 and 30? (a) 2 (b) 1 (c) 3 (d) 4
37. 37.What is the smallest prime factor of 65? (a) 13 (b) 5 (c) 1 (d) 2
38. 38.How many factors does 17 have? (a) 1 (b) 2 (c) 4 (d) 3
39. 39.How many factors does 90 have? (a) 11 (b) 12 (c) 13 (d) 10
40. 40.What is the smallest prime factor of 119? (a) 17 (b) 1 (c) 2 (d) 7

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