Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 23 · 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 9^15? (a) 1 (b) 8 (c) 9 (d) 7
2. 2.Which of these numbers is divisible by 3? (a) 2629 (b) 2631 (c) 2635 (d) 2632
3. 3.Which of these numbers is divisible by 8? (a) 1550 (b) 1538 (c) 1557 (d) 1544
4. 4.What is the remainder when 694 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
5. 5.What is the smallest number you must add to 113 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the smallest number you must add to 397 to make it divisible by 7? (a) 5 (b) 1 (c) 3 (d) 2
7. 7.Which of these numbers is divisible by 3? (a) 869 (b) 872 (c) 875 (d) 870
8. 8.Which of these numbers is divisible by 8? (a) 6156 (b) 6144 (c) 6139 (d) 6145
9. 9.What is the smallest number you must add to 183 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
10. 10.Which of these numbers is divisible by 6? (a) 2720 (b) 2712 (c) 2708 (d) 2714
11. 11.Which of these numbers is divisible by 11? (a) 5321 (b) 5325 (c) 5345 (d) 5324
12. 12.What is the remainder when 170 is divided by 3? (a) 2 (b) 1 (c) 0
13. 13.Which of these numbers is divisible by 3? (a) 2489 (b) 2482 (c) 2484 (d) 2485
14. 14.Which of these numbers is divisible by 8? (a) 393 (b) 405 (c) 392 (d) 388
15. 15.What is the smallest number you must add to 138 to make it divisible by 8? (a) 7 (b) 2 (c) 6 (d) 5
16. 16.Which of these numbers is divisible by 11? (a) 4601 (b) 4630 (c) 4613 (d) 4609
17. 17.What is the units digit of 8^47? (a) 4 (b) 6 (c) 8 (d) 2
18. 18.What is the units digit of 7^58? (a) 1 (b) 7 (c) 3 (d) 9
19. 19.What is the remainder when 876 is divided by 11? (a) 8 (b) 7 (c) 6 (d) 4
20. 20.What is the smallest number you must add to 227 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
21. 21.What is the remainder when 309 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
22. 22.Which of these numbers is divisible by 6? (a) 949 (b) 958 (c) 954 (d) 962
23. 23.What is the remainder when 85 is divided by 3? (a) 0 (b) 1 (c) 2
24. 24.What is the smallest number you must add to 117 to make it divisible by 4? (a) 2 (b) 3 (c) 1
25. 25.Which of these numbers is divisible by 4? (a) 2835 (b) 2837 (c) 2829 (d) 2832
26. 26.Which of these numbers is divisible by 6? (a) 3609 (b) 3605 (c) 3600 (d) 3595
27. 27.What is the smallest number you must add to 342 to make it divisible by 11? (a) 10 (b) 9 (c) 1 (d) 8
28. 28.Which of these numbers is divisible by 8? (a) 1248 (b) 1252 (c) 1245 (d) 1263
29. 29.Which of these numbers is divisible by 11? (a) 7205 (b) 7218 (c) 7197 (d) 7212
30. 30.What is the remainder when 374 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
31. 31.What is the units digit of 2^56? (a) 8 (b) 4 (c) 2 (d) 6
32. 32.Which of these numbers is divisible by 9? (a) 621 (b) 632 (c) 614 (d) 627
33. 33.What is the remainder when 287 is divided by 9? (a) 7 (b) 1 (c) 6 (d) 8
34. 34.Which of these numbers is divisible by 11? (a) 8199 (b) 8188 (c) 8211 (d) 8195
35. 35.What is the units digit of 3^30? (a) 7 (b) 9 (c) 3 (d) 1
36. 36.What is the units digit of 9^32? (a) 9 (b) 2 (c) 1 (d) 0
37. 37.Which of these numbers is divisible by 9? (a) 2050 (b) 2031 (c) 2034 (d) 2036
38. 38.Which of these numbers is divisible by 6? (a) 3588 (b) 3586 (c) 3589 (d) 3599
39. 39.What is the smallest number you must add to 173 to make it divisible by 11? (a) 3 (b) 8 (c) 4 (d) 2
40. 40.What is the units digit of 2^45? (a) 6 (b) 4 (c) 8 (d) 2

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