Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 405 · 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 285 to make it divisible by 9? (a) 6 (b) 3 (c) 2 (d) 4
2. 2.What is the smallest number you must add to 590 to make it divisible by 12? (a) 9 (b) 2 (c) 10 (d) 11
3. 3.Which of these numbers is divisible by 4? (a) 1102 (b) 1097 (c) 1096 (d) 1095
4. 4.What is the remainder when 249 is divided by 10? (a) 1 (b) 9 (c) 7 (d) 8
5. 5.What is the remainder when 620 is divided by 11? (a) 5 (b) 4 (c) 7 (d) 3
6. 6.Which of these numbers is divisible by 4? (a) 335 (b) 327 (c) 329 (d) 328
7. 7.Which of these numbers is divisible by 4? (a) 1239 (b) 1241 (c) 1236 (d) 1235
8. 8.What is the remainder when 136 is divided by 9? (a) 1 (b) 2 (c) 8 (d) 0
9. 9.What is the remainder when 164 is divided by 3? (a) 0 (b) 1 (c) 2
10. 10.What is the remainder when 255 is divided by 12? (a) 9 (b) 4 (c) 2 (d) 3
11. 11.What is the units digit of 7^40? (a) 9 (b) 3 (c) 7 (d) 1
12. 12.What is the units digit of 3^16? (a) 7 (b) 1 (c) 3 (d) 9
13. 13.What is the remainder when 251 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
14. 14.What is the smallest number you must add to 676 to make it divisible by 12? (a) 7 (b) 9 (c) 8 (d) 4
15. 15.Which of these numbers is divisible by 9? (a) 2871 (b) 2888 (c) 2868 (d) 2874
16. 16.What is the remainder when 81 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
17. 17.What is the smallest number you must add to 246 to make it divisible by 11? (a) 7 (b) 4 (c) 8 (d) 6
18. 18.What is the units digit of 4^24? (a) 4 (b) 5 (c) 7 (d) 6
19. 19.What is the smallest number you must add to 741 to make it divisible by 11? (a) 6 (b) 8 (c) 7 (d) 4
20. 20.What is the units digit of 9^20? (a) 9 (b) 2 (c) 1 (d) 0
21. 21.Which of these numbers is divisible by 8? (a) 4865 (b) 4863 (c) 4864 (d) 4879
22. 22.Which of these numbers is divisible by 4? (a) 3499 (b) 3506 (c) 3500 (d) 3503
23. 23.What is the units digit of 9^53? (a) 7 (b) 1 (c) 9 (d) 8
24. 24.What is the remainder when 111 is divided by 9? (a) 2 (b) 4 (c) 6 (d) 3
25. 25.What is the units digit of 9^21? (a) 9 (b) 1 (c) 7 (d) 8
26. 26.Which of these numbers is divisible by 8? (a) 4843 (b) 4840 (c) 4837 (d) 4855
27. 27.Which of these numbers is divisible by 6? (a) 2437 (b) 2452 (c) 2446 (d) 2442
28. 28.What is the smallest number you must add to 592 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
29. 29.Which of these numbers is divisible by 4? (a) 438 (b) 440 (c) 442 (d) 446
30. 30.What is the remainder when 127 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
31. 31.What is the units digit of 2^29? (a) 4 (b) 8 (c) 2 (d) 6
32. 32.What is the smallest number you must add to 213 to make it divisible by 4? (a) 2 (b) 3 (c) 1
33. 33.What is the remainder when 299 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
34. 34.Which of these numbers is divisible by 3? (a) 2321 (b) 2319 (c) 2324 (d) 2318
35. 35.What is the smallest number you must add to 239 to make it divisible by 4? (a) 1 (b) 2 (c) 3
36. 36.Which of these numbers is divisible by 11? (a) 1340 (b) 1324 (c) 1347 (d) 1331
37. 37.Which of these numbers is divisible by 4? (a) 3320 (b) 3326 (c) 3319 (d) 3322
38. 38.Which of these numbers is divisible by 9? (a) 7056 (b) 7055 (c) 7064 (d) 7069
39. 39.What is the remainder when 716 is divided by 9? (a) 5 (b) 7 (c) 4 (d) 6
40. 40.What is the remainder when 182 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2

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