Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 968 · 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 11? (a) 3729 (b) 3746 (c) 3737 (d) 3728
2. 2.Which of these numbers is divisible by 11? (a) 9408 (b) 9397 (c) 9392 (d) 9394
3. 3.What is the units digit of 2^57? (a) 2 (b) 8 (c) 4 (d) 6
4. 4.Which of these numbers is divisible by 8? (a) 2734 (b) 2720 (c) 2725 (d) 2716
5. 5.What is the smallest number you must add to 322 to make it divisible by 11? (a) 3 (b) 8 (c) 9 (d) 7
6. 6.Which of these numbers is divisible by 3? (a) 2219 (b) 2222 (c) 2217 (d) 2216
7. 7.What is the smallest number you must add to 408 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
8. 8.What is the units digit of 9^31? (a) 7 (b) 1 (c) 9 (d) 8
9. 9.Which of these numbers is divisible by 6? (a) 1725 (b) 1711 (c) 1721 (d) 1716
10. 10.What is the remainder when 215 is divided by 8? (a) 6 (b) 7 (c) 1 (d) 5
11. 11.What is the units digit of 8^27? (a) 4 (b) 2 (c) 8 (d) 6
12. 12.What is the remainder when 649 is divided by 10? (a) 8 (b) 9 (c) 1 (d) 7
13. 13.What is the units digit of 8^10? (a) 4 (b) 8 (c) 2 (d) 6
14. 14.What is the smallest number you must add to 98 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
15. 15.What is the smallest number you must add to 856 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
16. 16.Which of these numbers is divisible by 8? (a) 5304 (b) 5314 (c) 5303 (d) 5306
17. 17.What is the units digit of 8^36? (a) 2 (b) 8 (c) 6 (d) 4
18. 18.Which of these numbers is divisible by 6? (a) 4391 (b) 4386 (c) 4396 (d) 4385
19. 19.What is the smallest number you must add to 292 to make it divisible by 9? (a) 7 (b) 6 (c) 5 (d) 4
20. 20.What is the smallest number you must add to 165 to make it divisible by 12? (a) 2 (b) 4 (c) 3 (d) 9
21. 21.What is the remainder when 138 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
22. 22.Which of these numbers is divisible by 9? (a) 432 (b) 445 (c) 430 (d) 440
23. 23.Which of these numbers is divisible by 4? (a) 3218 (b) 3222 (c) 3226 (d) 3220
24. 24.What is the units digit of 2^37? (a) 4 (b) 6 (c) 8 (d) 2
25. 25.What is the units digit of 3^47? (a) 9 (b) 1 (c) 7 (d) 3
26. 26.What is the smallest number you must add to 321 to make it divisible by 12? (a) 4 (b) 2 (c) 3 (d) 9
27. 27.Which of these numbers is divisible by 6? (a) 3570 (b) 3575 (c) 3567 (d) 3581
28. 28.What is the remainder when 635 is divided by 8? (a) 2 (b) 4 (c) 3 (d) 5
29. 29.What is the units digit of 4^60? (a) 4 (b) 5 (c) 6 (d) 7
30. 30.What is the remainder when 202 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 463 (b) 466 (c) 470 (d) 465
32. 32.Which of these numbers is divisible by 3? (a) 1172 (b) 1168 (c) 1167 (d) 1165
33. 33.Which of these numbers is divisible by 6? (a) 5087 (b) 5081 (c) 5082 (d) 5093
34. 34.What is the smallest number you must add to 335 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 2
35. 35.What is the smallest number you must add to 238 to make it divisible by 9? (a) 5 (b) 7 (c) 4 (d) 6
36. 36.What is the remainder when 479 is divided by 8? (a) 5 (b) 6 (c) 1 (d) 7
37. 37.What is the remainder when 158 is divided by 9? (a) 4 (b) 5 (c) 7 (d) 6
38. 38.Which of these numbers is divisible by 4? (a) 1880 (b) 1883 (c) 1878 (d) 1886
39. 39.What is the smallest number you must add to 234 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 2
40. 40.Which of these numbers is divisible by 11? (a) 2124 (b) 2135 (c) 2134 (d) 2153

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