Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 660 · 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 3? (a) 1713 (b) 1714 (c) 1717 (d) 1711
2. 2.What is the units digit of 3^13? (a) 3 (b) 1 (c) 7 (d) 9
3. 3.Which of these numbers is divisible by 9? (a) 5392 (b) 5402 (c) 5391 (d) 5385
4. 4.What is the smallest number you must add to 557 to make it divisible by 7? (a) 2 (b) 4 (c) 3 (d) 5
5. 5.Which of these numbers is divisible by 3? (a) 1216 (b) 1214 (c) 1215 (d) 1219
6. 6.What is the remainder when 381 is divided by 7? (a) 3 (b) 2 (c) 4 (d) 5
7. 7.Which of these numbers is divisible by 3? (a) 468 (b) 466 (c) 472 (d) 469
8. 8.What is the smallest number you must add to 105 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 2
9. 9.Which of these numbers is divisible by 11? (a) 2686 (b) 2682 (c) 2673 (d) 2670
10. 10.What is the smallest number you must add to 932 to make it divisible by 12? (a) 5 (b) 4 (c) 3 (d) 8
11. 11.What is the remainder when 79 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
12. 12.What is the smallest number you must add to 155 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
13. 13.What is the remainder when 221 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
14. 14.What is the units digit of 2^58? (a) 6 (b) 8 (c) 2 (d) 4
15. 15.What is the remainder when 403 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
16. 16.Which of these numbers is divisible by 8? (a) 3669 (b) 3663 (c) 3664 (d) 3674
17. 17.What is the remainder when 581 is divided by 12? (a) 5 (b) 7 (c) 4 (d) 6
18. 18.What is the smallest number you must add to 775 to make it divisible by 10? (a) 5 (b) 4 (c) 7 (d) 6
19. 19.What is the units digit of 9^35? (a) 9 (b) 8 (c) 7 (d) 1
20. 20.What is the smallest number you must add to 113 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
21. 21.What is the smallest number you must add to 52 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the smallest number you must add to 328 to make it divisible by 11? (a) 9 (b) 1 (c) 2 (d) 3
23. 23.What is the units digit of 9^34? (a) 2 (b) 9 (c) 1 (d) 0
24. 24.Which of these numbers is divisible by 9? (a) 7956 (b) 7955 (c) 7963 (d) 7972
25. 25.What is the remainder when 184 is divided by 11? (a) 8 (b) 7 (c) 9 (d) 3
26. 26.What is the units digit of 2^22? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.What is the units digit of 2^11? (a) 4 (b) 6 (c) 8 (d) 2
28. 28.What is the units digit of 9^39? (a) 7 (b) 9 (c) 8 (d) 1
29. 29.What is the units digit of 3^21? (a) 7 (b) 9 (c) 1 (d) 3
30. 30.What is the units digit of 7^56? (a) 9 (b) 7 (c) 1 (d) 3
31. 31.What is the smallest number you must add to 547 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
32. 32.What is the remainder when 753 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
33. 33.What is the smallest number you must add to 610 to make it divisible by 8? (a) 5 (b) 2 (c) 6 (d) 7
34. 34.Which of these numbers is divisible by 9? (a) 1108 (b) 1107 (c) 1118 (d) 1102
35. 35.What is the remainder when 129 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
36. 36.What is the remainder when 164 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 3
37. 37.What is the remainder when 285 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
38. 38.What is the remainder when 83 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
39. 39.What is the smallest number you must add to 144 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 4
40. 40.What is the smallest number you must add to 584 to make it divisible by 9? (a) 3 (b) 2 (c) 8 (d) 1

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad