Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 175 · 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 169 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
2. 2.Which of these numbers is divisible by 8? (a) 1468 (b) 1456 (c) 1461 (d) 1452
3. 3.Which of these numbers is divisible by 8? (a) 6624 (b) 6626 (c) 6638 (d) 6623
4. 4.What is the units digit of 2^20? (a) 4 (b) 2 (c) 8 (d) 6
5. 5.What is the units digit of 8^58? (a) 8 (b) 2 (c) 4 (d) 6
6. 6.What is the units digit of 2^21? (a) 4 (b) 2 (c) 8 (d) 6
7. 7.What is the smallest number you must add to 206 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
8. 8.What is the remainder when 113 is divided by 11? (a) 2 (b) 3 (c) 4 (d) 8
9. 9.What is the remainder when 622 is divided by 10? (a) 2 (b) 1 (c) 8 (d) 3
10. 10.Which of these numbers is divisible by 3? (a) 2389 (b) 2387 (c) 2392 (d) 2388
11. 11.What is the remainder when 318 is divided by 11? (a) 9 (b) 1 (c) 8 (d) 10
12. 12.What is the smallest number you must add to 82 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the remainder when 734 is divided by 12? (a) 10 (b) 3 (c) 2 (d) 1
14. 14.What is the units digit of 7^32? (a) 9 (b) 7 (c) 1 (d) 3
15. 15.What is the units digit of 7^35? (a) 7 (b) 9 (c) 1 (d) 3
16. 16.Which of these numbers is divisible by 4? (a) 3407 (b) 3410 (c) 3415 (d) 3408
17. 17.Which of these numbers is divisible by 4? (a) 3589 (b) 3588 (c) 3594 (d) 3586
18. 18.What is the units digit of 2^17? (a) 4 (b) 8 (c) 2 (d) 6
19. 19.Which of these numbers is divisible by 3? (a) 2653 (b) 2651 (c) 2652 (d) 2656
20. 20.What is the units digit of 2^19? (a) 6 (b) 4 (c) 8 (d) 2
21. 21.Which of these numbers is divisible by 6? (a) 1355 (b) 1365 (c) 1360 (d) 1356
22. 22.What is the smallest number you must add to 216 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
23. 23.What is the units digit of 8^44? (a) 4 (b) 8 (c) 2 (d) 6
24. 24.What is the remainder when 113 is divided by 3? (a) 1 (b) 2 (c) 0
25. 25.Which of these numbers is divisible by 3? (a) 1849 (b) 1853 (c) 1848 (d) 1846
26. 26.Which of these numbers is divisible by 3? (a) 1084 (b) 1078 (c) 1080 (d) 1082
27. 27.What is the remainder when 102 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
28. 28.What is the smallest number you must add to 289 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 2
29. 29.What is the units digit of 7^53? (a) 3 (b) 7 (c) 1 (d) 9
30. 30.Which of these numbers is divisible by 8? (a) 2872 (b) 2886 (c) 2870 (d) 2877
31. 31.Which of these numbers is divisible by 8? (a) 2379 (b) 2370 (c) 2390 (d) 2376
32. 32.What is the smallest number you must add to 201 to make it divisible by 7? (a) 5 (b) 2 (c) 1 (d) 3
33. 33.What is the units digit of 7^59? (a) 1 (b) 9 (c) 3 (d) 7
34. 34.What is the smallest number you must add to 181 to make it divisible by 4? (a) 3 (b) 2 (c) 1
35. 35.What is the units digit of 3^10? (a) 3 (b) 9 (c) 7 (d) 1
36. 36.What is the units digit of 7^23? (a) 7 (b) 1 (c) 3 (d) 9
37. 37.Which of these numbers is divisible by 9? (a) 1665 (b) 1679 (c) 1668 (d) 1661
38. 38.Which of these numbers is divisible by 3? (a) 643 (b) 645 (c) 650 (d) 647
39. 39.Which of these numbers is divisible by 3? (a) 1437 (b) 1435 (c) 1441 (d) 1438
40. 40.Which of these numbers is divisible by 3? (a) 1360 (b) 1359 (c) 1358 (d) 1364

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