Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 540 · 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 units digit of 8^33? (a) 2 (b) 4 (c) 8 (d) 6
2. 2.What is the smallest number you must add to 865 to make it divisible by 11? (a) 5 (b) 4 (c) 7 (d) 3
3. 3.What is the units digit of 3^18? (a) 9 (b) 7 (c) 3 (d) 1
4. 4.What is the smallest number you must add to 223 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the units digit of 8^39? (a) 2 (b) 6 (c) 4 (d) 8
6. 6.What is the smallest number you must add to 416 to make it divisible by 12? (a) 3 (b) 8 (c) 5 (d) 4
7. 7.Which of these numbers is divisible by 9? (a) 7674 (b) 7668 (c) 7681 (d) 7661
8. 8.What is the remainder when 205 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 4
9. 9.What is the units digit of 2^51? (a) 6 (b) 4 (c) 2 (d) 8
10. 10.What is the units digit of 8^23? (a) 6 (b) 4 (c) 8 (d) 2
11. 11.What is the units digit of 9^12? (a) 2 (b) 9 (c) 1 (d) 0
12. 12.What is the smallest number you must add to 278 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 5
13. 13.What is the remainder when 46 is divided by 3? (a) 0 (b) 2 (c) 1
14. 14.Which of these numbers is divisible by 8? (a) 7008 (b) 7004 (c) 7021 (d) 7013
15. 15.What is the remainder when 205 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
16. 16.Which of these numbers is divisible by 4? (a) 248 (b) 255 (c) 246 (d) 251
17. 17.Which of these numbers is divisible by 8? (a) 4509 (b) 4514 (c) 4526 (d) 4512
18. 18.What is the remainder when 647 is divided by 10? (a) 3 (b) 8 (c) 7 (d) 6
19. 19.Which of these numbers is divisible by 11? (a) 1276 (b) 1297 (c) 1280 (d) 1273
20. 20.What is the units digit of 3^21? (a) 3 (b) 1 (c) 9 (d) 7
21. 21.Which of these numbers is divisible by 8? (a) 1541 (b) 1549 (c) 1536 (d) 1535
22. 22.Which of these numbers is divisible by 8? (a) 1004 (b) 1017 (c) 1012 (d) 1008
23. 23.Which of these numbers is divisible by 11? (a) 5325 (b) 5315 (c) 5313 (d) 5303
24. 24.Which of these numbers is divisible by 8? (a) 4841 (b) 4839 (c) 4832 (d) 4830
25. 25.Which of these numbers is divisible by 6? (a) 3414 (b) 3421 (c) 3409 (d) 3415
26. 26.Which of these numbers is divisible by 6? (a) 3459 (b) 3471 (c) 3463 (d) 3462
27. 27.Which of these numbers is divisible by 3? (a) 310 (b) 313 (c) 316 (d) 312
28. 28.What is the smallest number you must add to 253 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 2
29. 29.What is the smallest number you must add to 137 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 2
30. 30.What is the units digit of 7^17? (a) 1 (b) 9 (c) 3 (d) 7
31. 31.What is the smallest number you must add to 665 to make it divisible by 10? (a) 6 (b) 7 (c) 4 (d) 5
32. 32.What is the smallest number you must add to 256 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
33. 33.Which of these numbers is divisible by 9? (a) 7637 (b) 7643 (c) 7654 (d) 7641
34. 34.Which of these numbers is divisible by 8? (a) 3015 (b) 3008 (c) 3002 (d) 3018
35. 35.What is the remainder when 398 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
36. 36.What is the remainder when 110 is divided by 9? (a) 2 (b) 7 (c) 1 (d) 3
37. 37.Which of these numbers is divisible by 3? (a) 678 (b) 676 (c) 682 (d) 679
38. 38.What is the smallest number you must add to 221 to make it divisible by 4? (a) 1 (b) 2 (c) 3
39. 39.Which of these numbers is divisible by 11? (a) 7122 (b) 7131 (c) 7145 (d) 7128
40. 40.What is the smallest number you must add to 338 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2

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