Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 300 · 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.Which of these numbers is divisible by 8? (a) 1255 (b) 1240 (c) 1243 (d) 1236
2. 2.What is the remainder when 291 is divided by 9? (a) 2 (b) 4 (c) 3 (d) 6
3. 3.What is the remainder when 465 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
4. 4.What is the smallest number you must add to 184 to make it divisible by 3? (a) 1 (b) 2
5. 5.What is the smallest number you must add to 604 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 7
6. 6.Which of these numbers is divisible by 8? (a) 1584 (b) 1597 (c) 1590 (d) 1581
7. 7.What is the units digit of 3^28? (a) 7 (b) 3 (c) 1 (d) 9
8. 8.What is the remainder when 63 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
9. 9.Which of these numbers is divisible by 4? (a) 3443 (b) 3433 (c) 3436 (d) 3438
10. 10.Which of these numbers is divisible by 3? (a) 818 (b) 820 (c) 816 (d) 815
11. 11.What is the remainder when 238 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
12. 12.What is the remainder when 185 is divided by 12? (a) 4 (b) 6 (c) 5 (d) 7
13. 13.Which of these numbers is divisible by 11? (a) 4514 (b) 4508 (c) 4529 (d) 4510
14. 14.What is the remainder when 714 is divided by 9? (a) 4 (b) 2 (c) 6 (d) 3
15. 15.What is the smallest number you must add to 203 to make it divisible by 3? (a) 1 (b) 2
16. 16.Which of these numbers is divisible by 11? (a) 1022 (b) 991 (c) 1007 (d) 1001
17. 17.Which of these numbers is divisible by 6? (a) 261 (b) 265 (c) 264 (d) 275
18. 18.What is the remainder when 95 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
19. 19.Which of these numbers is divisible by 9? (a) 6505 (b) 6507 (c) 6517 (d) 6514
20. 20.What is the remainder when 695 is divided by 11? (a) 1 (b) 2 (c) 9 (d) 3
21. 21.What is the remainder when 139 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
22. 22.Which of these numbers is divisible by 9? (a) 6222 (b) 6210 (c) 6206 (d) 6217
23. 23.Which of these numbers is divisible by 11? (a) 8335 (b) 8338 (c) 8339 (d) 8356
24. 24.Which of these numbers is divisible by 4? (a) 271 (b) 272 (c) 279 (d) 273
25. 25.What is the units digit of 9^34? (a) 2 (b) 9 (c) 0 (d) 1
26. 26.Which of these numbers is divisible by 8? (a) 392 (b) 386 (c) 393 (d) 404
27. 27.What is the remainder when 112 is divided by 3? (a) 1 (b) 0 (c) 2
28. 28.What is the smallest number you must add to 512 to make it divisible by 7? (a) 1 (b) 5 (c) 6 (d) 4
29. 29.What is the smallest number you must add to 206 to make it divisible by 4? (a) 1 (b) 2 (c) 3
30. 30.What is the units digit of 3^42? (a) 3 (b) 9 (c) 1 (d) 7
31. 31.What is the smallest number you must add to 152 to make it divisible by 10? (a) 8 (b) 2 (c) 7 (d) 9
32. 32.What is the smallest number you must add to 327 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
33. 33.What is the smallest number you must add to 289 to make it divisible by 12? (a) 11 (b) 9 (c) 1 (d) 10
34. 34.What is the smallest number you must add to 530 to make it divisible by 11? (a) 2 (b) 8 (c) 9 (d) 10
35. 35.What is the units digit of 2^23? (a) 2 (b) 4 (c) 8 (d) 6
36. 36.What is the remainder when 98 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
37. 37.Which of these numbers is divisible by 6? (a) 4105 (b) 4120 (c) 4112 (d) 4110
38. 38.Which of these numbers is divisible by 6? (a) 3327 (b) 3318 (c) 3316 (d) 3321
39. 39.What is the remainder when 311 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
40. 40.Which of these numbers is divisible by 3? (a) 164 (b) 169 (c) 167 (d) 165

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