Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 576 · 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 347 is divided by 12? (a) 9 (b) 10 (c) 1 (d) 11
2. 2.What is the smallest number you must add to 316 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
3. 3.Which of these numbers is divisible by 8? (a) 5144 (b) 5142 (c) 5148 (d) 5158
4. 4.What is the remainder when 232 is divided by 3? (a) 2 (b) 1 (c) 0
5. 5.What is the units digit of 7^11? (a) 7 (b) 9 (c) 1 (d) 3
6. 6.What is the units digit of 8^53? (a) 6 (b) 2 (c) 8 (d) 4
7. 7.What is the remainder when 611 is divided by 8? (a) 4 (b) 2 (c) 3 (d) 5
8. 8.Which of these numbers is divisible by 11? (a) 4695 (b) 4698 (c) 4714 (d) 4697
9. 9.What is the remainder when 463 is divided by 11? (a) 0 (b) 1 (c) 10 (d) 2
10. 10.What is the units digit of 2^14? (a) 8 (b) 4 (c) 2 (d) 6
11. 11.What is the units digit of 2^11? (a) 6 (b) 2 (c) 4 (d) 8
12. 12.What is the smallest number you must add to 242 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
13. 13.Which of these numbers is divisible by 9? (a) 4823 (b) 4807 (c) 4815 (d) 4827
14. 14.What is the smallest number you must add to 253 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 1
15. 15.What is the remainder when 347 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
16. 16.What is the smallest number you must add to 105 to make it divisible by 4? (a) 2 (b) 1 (c) 3
17. 17.What is the smallest number you must add to 168 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1
18. 18.What is the remainder when 112 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
19. 19.Which of these numbers is divisible by 8? (a) 6174 (b) 6176 (c) 6177 (d) 6185
20. 20.Which of these numbers is divisible by 3? (a) 2188 (b) 2192 (c) 2190 (d) 2194
21. 21.What is the smallest number you must add to 275 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
22. 22.Which of these numbers is divisible by 3? (a) 1534 (b) 1528 (c) 1532 (d) 1530
23. 23.What is the remainder when 186 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
24. 24.What is the units digit of 3^44? (a) 3 (b) 7 (c) 1 (d) 9
25. 25.What is the units digit of 2^20? (a) 8 (b) 2 (c) 4 (d) 6
26. 26.Which of these numbers is divisible by 3? (a) 389 (b) 382 (c) 386 (d) 384
27. 27.Which of these numbers is divisible by 6? (a) 354 (b) 361 (c) 359 (d) 350
28. 28.What is the remainder when 846 is divided by 11? (a) 10 (b) 1 (c) 8 (d) 9
29. 29.What is the remainder when 602 is divided by 10? (a) 2 (b) 3 (c) 1 (d) 8
30. 30.What is the remainder when 268 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
31. 31.What is the smallest number you must add to 562 to make it divisible by 8? (a) 2 (b) 5 (c) 7 (d) 6
32. 32.What is the remainder when 164 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
33. 33.What is the remainder when 657 is divided by 11? (a) 3 (b) 7 (c) 8 (d) 9
34. 34.What is the units digit of 8^52? (a) 2 (b) 6 (c) 8 (d) 4
35. 35.What is the remainder when 193 is divided by 3? (a) 1 (b) 2 (c) 0
36. 36.What is the units digit of 3^59? (a) 1 (b) 7 (c) 9 (d) 3
37. 37.What is the remainder when 669 is divided by 10? (a) 9 (b) 1 (c) 8 (d) 7
38. 38.Which of these numbers is divisible by 11? (a) 5544 (b) 5559 (c) 5543 (d) 5545
39. 39.What is the remainder when 360 is divided by 7? (a) 2 (b) 5 (c) 3 (d) 4
40. 40.What is the units digit of 4^55? (a) 3 (b) 4 (c) 6 (d) 5

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