Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 61 · 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 4^13? (a) 4 (b) 5 (c) 6 (d) 3
2. 2.What is the remainder when 83 is divided by 8? (a) 5 (b) 2 (c) 4 (d) 3
3. 3.Which of these numbers is divisible by 3? (a) 800 (b) 806 (c) 803 (d) 801
4. 4.What is the units digit of 8^23? (a) 6 (b) 4 (c) 8 (d) 2
5. 5.What is the remainder when 209 is divided by 7? (a) 5 (b) 6 (c) 1 (d) 4
6. 6.What is the units digit of 7^57? (a) 7 (b) 3 (c) 1 (d) 9
7. 7.What is the units digit of 9^29? (a) 7 (b) 8 (c) 1 (d) 9
8. 8.What is the smallest number you must add to 272 to make it divisible by 7? (a) 1 (b) 6 (c) 3 (d) 2
9. 9.What is the units digit of 3^51? (a) 7 (b) 3 (c) 9 (d) 1
10. 10.What is the remainder when 107 is divided by 3? (a) 2 (b) 0 (c) 1
11. 11.What is the remainder when 241 is divided by 11? (a) 1 (b) 9 (c) 8 (d) 10
12. 12.What is the units digit of 2^28? (a) 4 (b) 6 (c) 8 (d) 2
13. 13.What is the units digit of 8^50? (a) 4 (b) 6 (c) 2 (d) 8
14. 14.What is the smallest number you must add to 161 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
15. 15.Which of these numbers is divisible by 11? (a) 7482 (b) 7469 (c) 7459 (d) 7470
16. 16.What is the remainder when 154 is divided by 10? (a) 5 (b) 4 (c) 6 (d) 3
17. 17.What is the units digit of 4^15? (a) 3 (b) 4 (c) 6 (d) 5
18. 18.Which of these numbers is divisible by 9? (a) 7883 (b) 7895 (c) 7884 (d) 7886
19. 19.What is the units digit of 7^49? (a) 3 (b) 7 (c) 1 (d) 9
20. 20.What is the smallest number you must add to 376 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 1
21. 21.Which of these numbers is divisible by 6? (a) 1394 (b) 1399 (c) 1388 (d) 1392
22. 22.What is the units digit of 8^53? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.Which of these numbers is divisible by 11? (a) 3380 (b) 3363 (c) 3371 (d) 3366
24. 24.What is the remainder when 287 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
25. 25.What is the smallest number you must add to 239 to make it divisible by 3? (a) 1 (b) 2
26. 26.Which of these numbers is divisible by 4? (a) 719 (b) 715 (c) 710 (d) 712
27. 27.Which of these numbers is divisible by 6? (a) 672 (b) 680 (c) 668 (d) 676
28. 28.What is the units digit of 8^29? (a) 8 (b) 4 (c) 2 (d) 6
29. 29.What is the units digit of 7^31? (a) 7 (b) 3 (c) 1 (d) 9
30. 30.Which of these numbers is divisible by 6? (a) 1078 (b) 1068 (c) 1069 (d) 1063
31. 31.What is the smallest number you must add to 222 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9
32. 32.Which of these numbers is divisible by 3? (a) 505 (b) 499 (c) 502 (d) 501
33. 33.What is the remainder when 723 is divided by 12? (a) 3 (b) 2 (c) 9 (d) 4
34. 34.What is the units digit of 3^36? (a) 1 (b) 7 (c) 3 (d) 9
35. 35.What is the remainder when 483 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
36. 36.Which of these numbers is divisible by 11? (a) 5446 (b) 5439 (c) 5433 (d) 5434
37. 37.What is the smallest number you must add to 112 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 8? (a) 4870 (b) 4875 (c) 4859 (d) 4864
39. 39.What is the remainder when 430 is divided by 11? (a) 10 (b) 1 (c) 2 (d) 0
40. 40.What is the remainder when 299 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 2

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