Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 542 · 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) 5666 (b) 5681 (c) 5665 (d) 5659
2. 2.Which of these numbers is divisible by 6? (a) 250 (b) 240 (c) 242 (d) 237
3. 3.What is the remainder when 245 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
4. 4.What is the remainder when 637 is divided by 12? (a) 11 (b) 1 (c) 2 (d) 0
5. 5.Which of these numbers is divisible by 6? (a) 3794 (b) 3786 (c) 3784 (d) 3787
6. 6.What is the remainder when 454 is divided by 7? (a) 6 (b) 1 (c) 5 (d) 4
7. 7.What is the units digit of 4^36? (a) 6 (b) 5 (c) 4 (d) 7
8. 8.Which of these numbers is divisible by 8? (a) 1274 (b) 1271 (c) 1284 (d) 1272
9. 9.What is the units digit of 8^44? (a) 8 (b) 2 (c) 6 (d) 4
10. 10.Which of these numbers is divisible by 3? (a) 1531 (b) 1529 (c) 1530 (d) 1534
11. 11.What is the remainder when 642 is divided by 11? (a) 4 (b) 7 (c) 5 (d) 3
12. 12.Which of these numbers is divisible by 3? (a) 197 (b) 194 (c) 192 (d) 191
13. 13.What is the smallest number you must add to 479 to make it divisible by 12? (a) 2 (b) 11 (c) 3 (d) 1
14. 14.What is the remainder when 668 is divided by 11? (a) 9 (b) 3 (c) 7 (d) 8
15. 15.What is the smallest number you must add to 256 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
16. 16.What is the units digit of 3^23? (a) 1 (b) 9 (c) 3 (d) 7
17. 17.Which of these numbers is divisible by 11? (a) 2975 (b) 3002 (c) 2981 (d) 2984
18. 18.What is the smallest number you must add to 454 to make it divisible by 12? (a) 1 (b) 2 (c) 3 (d) 10
19. 19.Which of these numbers is divisible by 4? (a) 1684 (b) 1686 (c) 1683 (d) 1691
20. 20.What is the smallest number you must add to 189 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
21. 21.What is the remainder when 887 is divided by 12? (a) 10 (b) 1 (c) 9 (d) 11
22. 22.What is the remainder when 125 is divided by 11? (a) 3 (b) 5 (c) 4 (d) 7
23. 23.What is the remainder when 145 is divided by 3? (a) 0 (b) 2 (c) 1
24. 24.What is the units digit of 7^14? (a) 7 (b) 1 (c) 3 (d) 9
25. 25.What is the units digit of 4^19? (a) 4 (b) 5 (c) 6 (d) 3
26. 26.What is the units digit of 9^23? (a) 8 (b) 1 (c) 9 (d) 7
27. 27.Which of these numbers is divisible by 3? (a) 608 (b) 614 (c) 609 (d) 611
28. 28.Which of these numbers is divisible by 8? (a) 627 (b) 618 (c) 615 (d) 616
29. 29.Which of these numbers is divisible by 8? (a) 4120 (b) 4132 (c) 4114 (d) 4126
30. 30.What is the smallest number you must add to 118 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 1
31. 31.What is the units digit of 7^37? (a) 3 (b) 1 (c) 7 (d) 9
32. 32.What is the units digit of 7^49? (a) 3 (b) 9 (c) 7 (d) 1
33. 33.What is the smallest number you must add to 164 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
34. 34.What is the remainder when 132 is divided by 9? (a) 7 (b) 3 (c) 6 (d) 5
35. 35.What is the units digit of 7^20? (a) 7 (b) 9 (c) 1 (d) 3
36. 36.What is the smallest number you must add to 214 to make it divisible by 4? (a) 3 (b) 1 (c) 2
37. 37.Which of these numbers is divisible by 6? (a) 608 (b) 605 (c) 606 (d) 615
38. 38.What is the remainder when 573 is divided by 12? (a) 10 (b) 3 (c) 8 (d) 9
39. 39.What is the remainder when 417 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
40. 40.What is the remainder when 94 is divided by 3? (a) 1 (b) 2 (c) 0

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