Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 635 · 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 smallest number you must add to 796 to make it divisible by 12? (a) 8 (b) 4 (c) 7 (d) 9
2. 2.What is the units digit of 2^31? (a) 2 (b) 4 (c) 8 (d) 6
3. 3.What is the units digit of 7^10? (a) 9 (b) 3 (c) 1 (d) 7
4. 4.Which of these numbers is divisible by 3? (a) 1207 (b) 1203 (c) 1201 (d) 1204
5. 5.What is the units digit of 8^11? (a) 2 (b) 8 (c) 4 (d) 6
6. 6.Which of these numbers is divisible by 3? (a) 615 (b) 619 (c) 613 (d) 617
7. 7.Which of these numbers is divisible by 3? (a) 1178 (b) 1173 (c) 1172 (d) 1175
8. 8.What is the smallest number you must add to 173 to make it divisible by 4? (a) 1 (b) 2 (c) 3
9. 9.What is the smallest number you must add to 94 to make it divisible by 8? (a) 6 (b) 1 (c) 3 (d) 2
10. 10.What is the units digit of 7^13? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.Which of these numbers is divisible by 3? (a) 1048 (b) 1047 (c) 1045 (d) 1052
12. 12.Which of these numbers is divisible by 9? (a) 6525 (b) 6524 (c) 6537 (d) 6533
13. 13.What is the remainder when 223 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
14. 14.What is the remainder when 302 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
15. 15.What is the units digit of 4^23? (a) 3 (b) 6 (c) 4 (d) 5
16. 16.What is the remainder when 611 is divided by 9? (a) 6 (b) 8 (c) 1 (d) 7
17. 17.What is the units digit of 8^17? (a) 8 (b) 6 (c) 2 (d) 4
18. 18.Which of these numbers is divisible by 11? (a) 4543 (b) 4546 (c) 4537 (d) 4563
19. 19.Which of these numbers is divisible by 6? (a) 5295 (b) 5299 (c) 5290 (d) 5292
20. 20.What is the units digit of 2^28? (a) 2 (b) 8 (c) 6 (d) 4
21. 21.What is the remainder when 198 is divided by 7? (a) 2 (b) 1 (c) 5 (d) 3
22. 22.Which of these numbers is divisible by 6? (a) 3158 (b) 3167 (c) 3155 (d) 3156
23. 23.What is the remainder when 659 is divided by 10? (a) 9 (b) 7 (c) 8 (d) 1
24. 24.Which of these numbers is divisible by 6? (a) 2074 (b) 2081 (c) 2084 (d) 2076
25. 25.What is the smallest number you must add to 680 to make it divisible by 12? (a) 8 (b) 4 (c) 5 (d) 3
26. 26.What is the remainder when 214 is divided by 3? (a) 0 (b) 1 (c) 2
27. 27.What is the remainder when 663 is divided by 11? (a) 4 (b) 8 (c) 3 (d) 2
28. 28.What is the remainder when 766 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
29. 29.Which of these numbers is divisible by 6? (a) 1772 (b) 1780 (c) 1776 (d) 1783
30. 30.What is the smallest number you must add to 532 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 6
31. 31.What is the smallest number you must add to 378 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 8
32. 32.What is the remainder when 640 is divided by 12? (a) 8 (b) 3 (c) 5 (d) 4
33. 33.What is the smallest number you must add to 492 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4
34. 34.What is the units digit of 7^57? (a) 9 (b) 7 (c) 1 (d) 3
35. 35.What is the remainder when 97 is divided by 3? (a) 1 (b) 2 (c) 0
36. 36.What is the remainder when 217 is divided by 9? (a) 2 (b) 1 (c) 0 (d) 8
37. 37.Which of these numbers is divisible by 6? (a) 5378 (b) 5382 (c) 5387 (d) 5393
38. 38.What is the units digit of 3^12? (a) 9 (b) 7 (c) 3 (d) 1
39. 39.What is the smallest number you must add to 704 to make it divisible by 9? (a) 2 (b) 6 (c) 7 (d) 8
40. 40.Which of these numbers is divisible by 6? (a) 3716 (b) 3714 (c) 3711 (d) 3722

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