Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 947 · 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 331 is divided by 8? (a) 2 (b) 5 (c) 3 (d) 4
2. 2.What is the smallest number you must add to 612 to make it divisible by 11? (a) 5 (b) 4 (c) 3 (d) 7
3. 3.What is the smallest number you must add to 182 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
4. 4.What is the remainder when 793 is divided by 12? (a) 2 (b) 11 (c) 0 (d) 1
5. 5.Which of these numbers is divisible by 6? (a) 4803 (b) 4799 (c) 4794 (d) 4789
6. 6.What is the smallest number you must add to 230 to make it divisible by 7? (a) 3 (b) 6 (c) 1 (d) 2
7. 7.What is the units digit of 8^50? (a) 8 (b) 2 (c) 6 (d) 4
8. 8.What is the remainder when 163 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
9. 9.Which of these numbers is divisible by 3? (a) 2405 (b) 2407 (c) 2403 (d) 2401
10. 10.What is the remainder when 461 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
11. 11.What is the smallest number you must add to 168 to make it divisible by 9? (a) 3 (b) 2 (c) 6 (d) 4
12. 12.What is the smallest number you must add to 395 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
13. 13.Which of these numbers is divisible by 9? (a) 531 (b) 532 (c) 525 (d) 541
14. 14.What is the remainder when 305 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
15. 15.Which of these numbers is divisible by 4? (a) 3153 (b) 3147 (c) 3148 (d) 3149
16. 16.Which of these numbers is divisible by 8? (a) 1702 (b) 1688 (c) 1684 (d) 1692
17. 17.Which of these numbers is divisible by 6? (a) 273 (b) 287 (c) 276 (d) 279
18. 18.Which of these numbers is divisible by 3? (a) 1851 (b) 1852 (c) 1849 (d) 1855
19. 19.What is the remainder when 459 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 3
20. 20.Which of these numbers is divisible by 8? (a) 3272 (b) 3276 (c) 3265 (d) 3281
21. 21.Which of these numbers is divisible by 8? (a) 2952 (b) 2961 (c) 2959 (d) 2951
22. 22.Which of these numbers is divisible by 8? (a) 5940 (b) 5946 (c) 5944 (d) 5959
23. 23.Which of these numbers is divisible by 6? (a) 1832 (b) 1837 (c) 1845 (d) 1836
24. 24.What is the units digit of 7^27? (a) 9 (b) 7 (c) 3 (d) 1
25. 25.Which of these numbers is divisible by 3? (a) 614 (b) 610 (c) 617 (d) 612
26. 26.What is the smallest number you must add to 164 to make it divisible by 12? (a) 8 (b) 5 (c) 4 (d) 3
27. 27.Which of these numbers is divisible by 8? (a) 408 (b) 405 (c) 409 (d) 419
28. 28.What is the remainder when 317 is divided by 12? (a) 7 (b) 4 (c) 5 (d) 6
29. 29.What is the smallest number you must add to 240 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 2
30. 30.What is the remainder when 151 is divided by 6? (a) 0 (b) 5 (c) 1 (d) 2
31. 31.What is the remainder when 423 is divided by 11? (a) 6 (b) 5 (c) 7 (d) 4
32. 32.What is the remainder when 77 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1
33. 33.What is the remainder when 244 is divided by 9? (a) 1 (b) 0 (c) 2 (d) 8
34. 34.What is the units digit of 3^30? (a) 3 (b) 9 (c) 1 (d) 7
35. 35.What is the remainder when 504 is divided by 11? (a) 2 (b) 8 (c) 10 (d) 9
36. 36.What is the remainder when 525 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
37. 37.Which of these numbers is divisible by 8? (a) 1231 (b) 1232 (c) 1238 (d) 1243
38. 38.Which of these numbers is divisible by 4? (a) 825 (b) 824 (c) 823 (d) 829
39. 39.What is the remainder when 76 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 0
40. 40.What is the remainder when 319 is divided by 7? (a) 3 (b) 5 (c) 6 (d) 4

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