Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 255 · 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 281 is divided by 12? (a) 7 (b) 5 (c) 4 (d) 6
2. 2.Which of these numbers is divisible by 4? (a) 840 (b) 845 (c) 841 (d) 839
3. 3.What is the remainder when 368 is divided by 10? (a) 8 (b) 7 (c) 2 (d) 9
4. 4.Which of these numbers is divisible by 11? (a) 9792 (b) 9780 (c) 9790 (d) 9802
5. 5.Which of these numbers is divisible by 4? (a) 695 (b) 689 (c) 697 (d) 692
6. 6.What is the remainder when 85 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
7. 7.What is the units digit of 8^60? (a) 4 (b) 2 (c) 8 (d) 6
8. 8.What is the smallest number you must add to 268 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
9. 9.What is the smallest number you must add to 627 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
10. 10.Which of these numbers is divisible by 8? (a) 357 (b) 361 (c) 375 (d) 360
11. 11.What is the smallest number you must add to 225 to make it divisible by 4? (a) 2 (b) 3 (c) 1
12. 12.What is the remainder when 231 is divided by 10? (a) 2 (b) 9 (c) 0 (d) 1
13. 13.What is the units digit of 7^47? (a) 1 (b) 7 (c) 9 (d) 3
14. 14.What is the units digit of 2^14? (a) 4 (b) 6 (c) 8 (d) 2
15. 15.What is the units digit of 3^46? (a) 3 (b) 9 (c) 1 (d) 7
16. 16.Which of these numbers is divisible by 9? (a) 4714 (b) 4698 (c) 4696 (d) 4700
17. 17.Which of these numbers is divisible by 4? (a) 3155 (b) 3162 (c) 3156 (d) 3158
18. 18.Which of these numbers is divisible by 3? (a) 901 (b) 904 (c) 908 (d) 903
19. 19.What is the remainder when 290 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
20. 20.What is the remainder when 178 is divided by 3? (a) 1 (b) 2 (c) 0
21. 21.What is the smallest number you must add to 525 to make it divisible by 10? (a) 7 (b) 4 (c) 6 (d) 5
22. 22.Which of these numbers is divisible by 4? (a) 425 (b) 430 (c) 421 (d) 424
23. 23.What is the smallest number you must add to 829 to make it divisible by 11? (a) 6 (b) 4 (c) 7 (d) 8
24. 24.What is the smallest number you must add to 717 to make it divisible by 11? (a) 10 (b) 2 (c) 9 (d) 8
25. 25.Which of these numbers is divisible by 3? (a) 595 (b) 601 (c) 597 (d) 599
26. 26.What is the units digit of 8^52? (a) 6 (b) 2 (c) 8 (d) 4
27. 27.What is the remainder when 274 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
28. 28.What is the remainder when 136 is divided by 3? (a) 0 (b) 2 (c) 1
29. 29.What is the smallest number you must add to 293 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
30. 30.What is the remainder when 623 is divided by 11? (a) 8 (b) 7 (c) 4 (d) 6
31. 31.Which of these numbers is divisible by 8? (a) 2831 (b) 2847 (c) 2832 (d) 2835
32. 32.What is the remainder when 56 is divided by 5? (a) 1 (b) 0 (c) 4 (d) 2
33. 33.What is the units digit of 8^21? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.What is the remainder when 199 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
35. 35.What is the smallest number you must add to 601 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 1
36. 36.Which of these numbers is divisible by 6? (a) 5205 (b) 5208 (c) 5209 (d) 5219
37. 37.What is the smallest number you must add to 338 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1
38. 38.Which of these numbers is divisible by 11? (a) 2902 (b) 2913 (c) 2922 (d) 2904
39. 39.What is the remainder when 572 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
40. 40.What is the remainder when 263 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3

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