Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 285 · 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 310 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
2. 2.What is the units digit of 9^27? (a) 1 (b) 7 (c) 8 (d) 9
3. 3.What is the remainder when 43 is divided by 3? (a) 0 (b) 2 (c) 1
4. 4.What is the units digit of 9^59? (a) 9 (b) 8 (c) 7 (d) 1
5. 5.What is the units digit of 2^20? (a) 4 (b) 8 (c) 6 (d) 2
6. 6.What is the units digit of 2^56? (a) 6 (b) 8 (c) 2 (d) 4
7. 7.What is the smallest number you must add to 799 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 9
8. 8.What is the remainder when 47 is divided by 3? (a) 2 (b) 0 (c) 1
9. 9.What is the units digit of 8^28? (a) 6 (b) 4 (c) 2 (d) 8
10. 10.What is the smallest number you must add to 328 to make it divisible by 9? (a) 4 (b) 7 (c) 5 (d) 6
11. 11.What is the smallest number you must add to 291 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
12. 12.What is the remainder when 832 is divided by 11? (a) 8 (b) 7 (c) 6 (d) 4
13. 13.What is the remainder when 290 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 4
14. 14.Which of these numbers is divisible by 8? (a) 1211 (b) 1195 (c) 1202 (d) 1200
15. 15.What is the remainder when 609 is divided by 12? (a) 8 (b) 3 (c) 10 (d) 9
16. 16.Which of these numbers is divisible by 11? (a) 745 (b) 729 (c) 720 (d) 726
17. 17.What is the units digit of 3^14? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.Which of these numbers is divisible by 6? (a) 2541 (b) 2535 (c) 2538 (d) 2548
19. 19.What is the units digit of 9^29? (a) 8 (b) 1 (c) 9 (d) 7
20. 20.What is the smallest number you must add to 268 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
21. 21.Which of these numbers is divisible by 9? (a) 2885 (b) 2863 (c) 2871 (d) 2879
22. 22.What is the smallest number you must add to 862 to make it divisible by 12? (a) 3 (b) 1 (c) 10 (d) 2
23. 23.What is the remainder when 262 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
24. 24.What is the units digit of 3^18? (a) 7 (b) 3 (c) 9 (d) 1
25. 25.What is the units digit of 9^60? (a) 9 (b) 1 (c) 0 (d) 2
26. 26.What is the smallest number you must add to 834 to make it divisible by 12? (a) 5 (b) 6 (c) 8 (d) 7
27. 27.What is the units digit of 9^23? (a) 1 (b) 7 (c) 9 (d) 8
28. 28.What is the remainder when 210 is divided by 11? (a) 2 (b) 0 (c) 10 (d) 1
29. 29.Which of these numbers is divisible by 3? (a) 2426 (b) 2421 (c) 2419 (d) 2423
30. 30.What is the remainder when 279 is divided by 12? (a) 3 (b) 4 (c) 2 (d) 9
31. 31.What is the smallest number you must add to 235 to make it divisible by 3? (a) 1 (b) 2
32. 32.Which of these numbers is divisible by 4? (a) 1369 (b) 1374 (c) 1367 (d) 1368
33. 33.Which of these numbers is divisible by 4? (a) 765 (b) 759 (c) 760 (d) 762
34. 34.What is the remainder when 452 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
35. 35.Which of these numbers is divisible by 6? (a) 2224 (b) 2209 (c) 2214 (d) 2219
36. 36.Which of these numbers is divisible by 4? (a) 3185 (b) 3182 (c) 3177 (d) 3180
37. 37.What is the smallest number you must add to 81 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 4
38. 38.Which of these numbers is divisible by 9? (a) 4095 (b) 4111 (c) 4098 (d) 4087
39. 39.What is the smallest number you must add to 135 to make it divisible by 11? (a) 7 (b) 8 (c) 9 (d) 3
40. 40.Which of these numbers is divisible by 6? (a) 4523 (b) 4518 (c) 4526 (d) 4514

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