Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 600 · 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 200 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 3
2. 2.What is the smallest number you must add to 82 to make it divisible by 3? (a) 2 (b) 1
3. 3.Which of these numbers is divisible by 6? (a) 3674 (b) 3686 (c) 3679 (d) 3678
4. 4.Which of these numbers is divisible by 4? (a) 209 (b) 204 (c) 201 (d) 206
5. 5.What is the remainder when 106 is divided by 3? (a) 1 (b) 0 (c) 2
6. 6.What is the remainder when 118 is divided by 11? (a) 8 (b) 3 (c) 9 (d) 7
7. 7.What is the smallest number you must add to 109 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the units digit of 4^21? (a) 6 (b) 5 (c) 3 (d) 4
9. 9.What is the smallest number you must add to 211 to make it divisible by 4? (a) 3 (b) 2 (c) 1
10. 10.Which of these numbers is divisible by 6? (a) 456 (b) 453 (c) 467 (d) 460
11. 11.What is the units digit of 7^24? (a) 3 (b) 1 (c) 7 (d) 9
12. 12.What is the smallest number you must add to 84 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
13. 13.What is the remainder when 536 is divided by 7? (a) 5 (b) 3 (c) 6 (d) 4
14. 14.What is the units digit of 9^49? (a) 8 (b) 7 (c) 1 (d) 9
15. 15.What is the remainder when 245 is divided by 10? (a) 5 (b) 4 (c) 6 (d) 7
16. 16.What is the units digit of 2^53? (a) 2 (b) 6 (c) 8 (d) 4
17. 17.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
18. 18.Which of these numbers is divisible by 8? (a) 1045 (b) 1062 (c) 1049 (d) 1048
19. 19.Which of these numbers is divisible by 11? (a) 1583 (b) 1600 (c) 1589 (d) 1584
20. 20.Which of these numbers is divisible by 6? (a) 4634 (b) 4643 (c) 4631 (d) 4632
21. 21.Which of these numbers is divisible by 6? (a) 1033 (b) 1031 (c) 1032 (d) 1039
22. 22.What is the smallest number you must add to 691 to make it divisible by 11? (a) 2 (b) 3 (c) 1 (d) 9
23. 23.What is the smallest number you must add to 111 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
24. 24.What is the smallest number you must add to 278 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
25. 25.Which of these numbers is divisible by 11? (a) 737 (b) 754 (c) 728 (d) 744
26. 26.What is the units digit of 4^29? (a) 6 (b) 3 (c) 5 (d) 4
27. 27.What is the smallest number you must add to 206 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
28. 28.What is the smallest number you must add to 143 to make it divisible by 3? (a) 2 (b) 1
29. 29.Which of these numbers is divisible by 9? (a) 1650 (b) 1660 (c) 1647 (d) 1646
30. 30.What is the smallest number you must add to 61 to make it divisible by 3? (a) 2 (b) 1
31. 31.What is the units digit of 9^29? (a) 1 (b) 7 (c) 8 (d) 9
32. 32.Which of these numbers is divisible by 4? (a) 2031 (b) 2026 (c) 2021 (d) 2024
33. 33.Which of these numbers is divisible by 8? (a) 378 (b) 364 (c) 368 (d) 371
34. 34.What is the units digit of 3^10? (a) 9 (b) 7 (c) 1 (d) 3
35. 35.What is the remainder when 346 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 4
36. 36.What is the remainder when 137 is divided by 7? (a) 6 (b) 4 (c) 3 (d) 5
37. 37.Which of these numbers is divisible by 11? (a) 1133 (b) 1145 (c) 1143 (d) 1126
38. 38.What is the smallest number you must add to 154 to make it divisible by 4? (a) 1 (b) 2 (c) 3
39. 39.What is the remainder when 357 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
40. 40.What is the remainder when 663 is divided by 12? (a) 4 (b) 9 (c) 3 (d) 2

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