Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 720 · 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 smallest number you must add to 536 to make it divisible by 11? (a) 4 (b) 8 (c) 2 (d) 3
2. 2.What is the remainder when 547 is divided by 8? (a) 3 (b) 2 (c) 5 (d) 4
3. 3.What is the smallest number you must add to 185 to make it divisible by 12? (a) 5 (b) 7 (c) 8 (d) 6
4. 4.What is the smallest number you must add to 482 to make it divisible by 7? (a) 6 (b) 2 (c) 3 (d) 1
5. 5.What is the smallest number you must add to 452 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 3
6. 6.Which of these numbers is divisible by 4? (a) 224 (b) 229 (c) 226 (d) 222
7. 7.What is the units digit of 7^23? (a) 1 (b) 3 (c) 7 (d) 9
8. 8.Which of these numbers is divisible by 4? (a) 3387 (b) 3394 (c) 3388 (d) 3389
9. 9.What is the smallest number you must add to 376 to make it divisible by 9? (a) 7 (b) 2 (c) 3 (d) 1
10. 10.What is the units digit of 2^52? (a) 4 (b) 6 (c) 2 (d) 8
11. 11.What is the smallest number you must add to 106 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
12. 12.What is the units digit of 4^55? (a) 3 (b) 5 (c) 6 (d) 4
13. 13.Which of these numbers is divisible by 4? (a) 684 (b) 691 (c) 682 (d) 687
14. 14.What is the smallest number you must add to 260 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
15. 15.What is the units digit of 2^58? (a) 8 (b) 4 (c) 6 (d) 2
16. 16.What is the units digit of 9^29? (a) 8 (b) 7 (c) 9 (d) 1
17. 17.What is the remainder when 579 is divided by 12? (a) 9 (b) 2 (c) 4 (d) 3
18. 18.What is the remainder when 229 is divided by 3? (a) 0 (b) 1 (c) 2
19. 19.Which of these numbers is divisible by 3? (a) 468 (b) 469 (c) 473 (d) 467
20. 20.Which of these numbers is divisible by 4? (a) 3545 (b) 3539 (c) 3540 (d) 3542
21. 21.Which of these numbers is divisible by 8? (a) 2402 (b) 2418 (c) 2415 (d) 2408
22. 22.Which of these numbers is divisible by 3? (a) 2307 (b) 2312 (c) 2309 (d) 2305
23. 23.Which of these numbers is divisible by 11? (a) 2683 (b) 2662 (c) 2670 (d) 2660
24. 24.What is the remainder when 842 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 8
25. 25.Which of these numbers is divisible by 4? (a) 2719 (b) 2720 (c) 2726 (d) 2723
26. 26.What is the remainder when 37 is divided by 3? (a) 0 (b) 2 (c) 1
27. 27.What is the units digit of 2^29? (a) 4 (b) 6 (c) 2 (d) 8
28. 28.What is the smallest number you must add to 198 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
29. 29.What is the remainder when 100 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
30. 30.What is the remainder when 433 is divided by 8? (a) 0 (b) 1 (c) 7 (d) 2
31. 31.What is the remainder when 276 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
32. 32.Which of these numbers is divisible by 4? (a) 1627 (b) 1629 (c) 1628 (d) 1635
33. 33.Which of these numbers is divisible by 8? (a) 439 (b) 427 (c) 446 (d) 432
34. 34.Which of these numbers is divisible by 4? (a) 3176 (b) 3175 (c) 3183 (d) 3178
35. 35.What is the smallest number you must add to 135 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
36. 36.Which of these numbers is divisible by 6? (a) 3482 (b) 3478 (c) 3470 (d) 3474
37. 37.What is the smallest number you must add to 66 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
38. 38.What is the units digit of 2^38? (a) 4 (b) 2 (c) 6 (d) 8
39. 39.Which of these numbers is divisible by 11? (a) 2519 (b) 2521 (c) 2539 (d) 2509
40. 40.Which of these numbers is divisible by 4? (a) 872 (b) 869 (c) 873 (d) 878

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