Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 574 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.What is the remainder when 304 is divided by 10? (a) 6 (b) 3 (c) 5 (d) 4
2. 2.What is the remainder when 207 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
3. 3.What is the smallest number you must add to 583 to make it divisible by 10? (a) 7 (b) 3 (c) 8 (d) 6
4. 4.What is the remainder when 454 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
5. 5.Which of these numbers is divisible by 8? (a) 3714 (b) 3727 (c) 3733 (d) 3720
6. 6.Which of these numbers is divisible by 9? (a) 7413 (b) 7416 (c) 7420 (d) 7429
7. 7.Which of these numbers is divisible by 3? (a) 140 (b) 141 (c) 146 (d) 143
8. 8.Which of these numbers is divisible by 4? (a) 3119 (b) 3125 (c) 3120 (d) 3122
9. 9.Which of these numbers is divisible by 6? (a) 647 (b) 648 (c) 656 (d) 649
10. 10.What is the units digit of 9^53? (a) 1 (b) 8 (c) 9 (d) 7
11. 11.What is the remainder when 522 is divided by 11? (a) 5 (b) 4 (c) 7 (d) 6
12. 12.Which of these numbers is divisible by 8? (a) 416 (b) 412 (c) 422 (d) 431
13. 13.Which of these numbers is divisible by 11? (a) 1819 (b) 1804 (c) 1796 (d) 1806
14. 14.What is the units digit of 7^56? (a) 7 (b) 1 (c) 9 (d) 3
15. 15.What is the remainder when 632 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
16. 16.Which of these numbers is divisible by 3? (a) 619 (b) 618 (c) 616 (d) 623
17. 17.What is the smallest number you must add to 215 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3
18. 18.What is the remainder when 204 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
19. 19.What is the smallest number you must add to 338 to make it divisible by 10? (a) 8 (b) 1 (c) 2 (d) 3
20. 20.What is the units digit of 9^32? (a) 0 (b) 2 (c) 9 (d) 1
21. 21.Which of these numbers is divisible by 9? (a) 5718 (b) 5715 (c) 5707 (d) 5725
22. 22.What is the smallest number you must add to 185 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
23. 23.What is the units digit of 9^60? (a) 9 (b) 2 (c) 1 (d) 0
24. 24.What is the units digit of 4^24? (a) 5 (b) 7 (c) 6 (d) 4
25. 25.What is the units digit of 8^12? (a) 8 (b) 6 (c) 2 (d) 4
26. 26.What is the remainder when 198 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 2506 (b) 2504 (c) 2505 (d) 2510
28. 28.What is the remainder when 589 is divided by 11? (a) 7 (b) 5 (c) 6 (d) 8
29. 29.Which of these numbers is divisible by 6? (a) 564 (b) 561 (c) 571 (d) 566
30. 30.Which of these numbers is divisible by 6? (a) 2730 (b) 2731 (c) 2739 (d) 2727
31. 31.What is the remainder when 529 is divided by 9? (a) 8 (b) 2 (c) 7 (d) 6
32. 32.Which of these numbers is divisible by 6? (a) 4649 (b) 4654 (c) 4644 (d) 4639
33. 33.What is the smallest number you must add to 205 to make it divisible by 7? (a) 2 (b) 5 (c) 6 (d) 4
34. 34.Which of these numbers is divisible by 3? (a) 1139 (b) 1140 (c) 1142 (d) 1145
35. 35.Which of these numbers is divisible by 6? (a) 1147 (b) 1136 (c) 1140 (d) 1143
36. 36.What is the smallest number you must add to 128 to make it divisible by 3? (a) 1 (b) 2
37. 37.Which of these numbers is divisible by 6? (a) 4653 (b) 4659 (c) 4665 (d) 4656
38. 38.What is the remainder when 491 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 4
39. 39.Which of these numbers is divisible by 3? (a) 154 (b) 157 (c) 156 (d) 161
40. 40.What is the units digit of 4^28? (a) 5 (b) 7 (c) 4 (d) 6

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