Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 520 · 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) 7018 (b) 7016 (c) 7026 (d) 7014
2. 2.Which of these numbers is divisible by 3? (a) 2533 (b) 2537 (c) 2531 (d) 2532
3. 3.What is the remainder when 573 is divided by 8? (a) 4 (b) 5 (c) 6 (d) 3
4. 4.Which of these numbers is divisible by 6? (a) 2832 (b) 2837 (c) 2841 (d) 2828
5. 5.Which of these numbers is divisible by 11? (a) 2505 (b) 2508 (c) 2522 (d) 2513
6. 6.What is the remainder when 459 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
7. 7.Which of these numbers is divisible by 3? (a) 824 (b) 819 (c) 820 (d) 818
8. 8.What is the smallest number you must add to 339 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
9. 9.What is the smallest number you must add to 118 to make it divisible by 4? (a) 2 (b) 3 (c) 1
10. 10.What is the remainder when 122 is divided by 9? (a) 4 (b) 5 (c) 7 (d) 6
11. 11.What is the smallest number you must add to 706 to make it divisible by 11? (a) 2 (b) 10 (c) 9 (d) 8
12. 12.What is the units digit of 9^35? (a) 9 (b) 1 (c) 7 (d) 8
13. 13.What is the remainder when 137 is divided by 3? (a) 0 (b) 2 (c) 1
14. 14.Which of these numbers is divisible by 9? (a) 4273 (b) 4280 (c) 4289 (d) 4275
15. 15.Which of these numbers is divisible by 6? (a) 4416 (b) 4418 (c) 4424 (d) 4415
16. 16.Which of these numbers is divisible by 3? (a) 1079 (b) 1081 (c) 1085 (d) 1080
17. 17.What is the units digit of 3^53? (a) 7 (b) 1 (c) 3 (d) 9
18. 18.What is the remainder when 239 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
19. 19.What is the smallest number you must add to 485 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
20. 20.What is the units digit of 2^33? (a) 6 (b) 2 (c) 4 (d) 8
21. 21.Which of these numbers is divisible by 11? (a) 8393 (b) 8389 (c) 8401 (d) 8414
22. 22.What is the smallest number you must add to 101 to make it divisible by 4? (a) 3 (b) 2 (c) 1
23. 23.What is the smallest number you must add to 87 to make it divisible by 8? (a) 3 (b) 7 (c) 2 (d) 1
24. 24.What is the remainder when 357 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
25. 25.Which of these numbers is divisible by 11? (a) 4798 (b) 4807 (c) 4821 (d) 4816
26. 26.What is the remainder when 458 is divided by 10? (a) 2 (b) 9 (c) 8 (d) 7
27. 27.What is the units digit of 9^23? (a) 8 (b) 7 (c) 9 (d) 1
28. 28.Which of these numbers is divisible by 3? (a) 1938 (b) 1937 (c) 1943 (d) 1939
29. 29.Which of these numbers is divisible by 11? (a) 3136 (b) 3115 (c) 3130 (d) 3124
30. 30.Which of these numbers is divisible by 3? (a) 2308 (b) 2306 (c) 2312 (d) 2307
31. 31.What is the remainder when 100 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.Which of these numbers is divisible by 4? (a) 3480 (b) 3487 (c) 3478 (d) 3482
33. 33.What is the remainder when 212 is divided by 3? (a) 0 (b) 1 (c) 2
34. 34.What is the remainder when 381 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 0
35. 35.What is the remainder when 237 is divided by 11? (a) 5 (b) 6 (c) 8 (d) 7
36. 36.Which of these numbers is divisible by 8? (a) 1230 (b) 1216 (c) 1217 (d) 1209
37. 37.What is the units digit of 4^19? (a) 6 (b) 4 (c) 3 (d) 5
38. 38.What is the remainder when 239 is divided by 8? (a) 5 (b) 7 (c) 1 (d) 6
39. 39.What is the remainder when 391 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
40. 40.What is the remainder when 447 is divided by 11? (a) 7 (b) 8 (c) 4 (d) 6

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