Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 244 · 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 remainder when 82 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
2. 2.Which of these numbers is divisible by 11? (a) 9746 (b) 9766 (c) 9741 (d) 9753
3. 3.What is the smallest number you must add to 192 to make it divisible by 10? (a) 2 (b) 9 (c) 8 (d) 7
4. 4.Which of these numbers is divisible by 6? (a) 860 (b) 849 (c) 852 (d) 857
5. 5.What is the smallest number you must add to 277 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
6. 6.Which of these numbers is divisible by 3? (a) 2668 (b) 2670 (c) 2674 (d) 2672
7. 7.Which of these numbers is divisible by 9? (a) 5982 (b) 5976 (c) 5975 (d) 5987
8. 8.Which of these numbers is divisible by 3? (a) 2487 (b) 2485 (c) 2489 (d) 2491
9. 9.What is the remainder when 142 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
10. 10.Which of these numbers is divisible by 4? (a) 2730 (b) 2724 (c) 2727 (d) 2721
11. 11.Which of these numbers is divisible by 8? (a) 889 (b) 897 (c) 896 (d) 905
12. 12.What is the smallest number you must add to 100 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the units digit of 4^38? (a) 4 (b) 6 (c) 7 (d) 5
14. 14.What is the units digit of 3^20? (a) 3 (b) 7 (c) 1 (d) 9
15. 15.What is the units digit of 7^42? (a) 3 (b) 1 (c) 9 (d) 7
16. 16.What is the smallest number you must add to 230 to make it divisible by 4? (a) 3 (b) 1 (c) 2
17. 17.Which of these numbers is divisible by 6? (a) 5194 (b) 5186 (c) 5200 (d) 5190
18. 18.What is the units digit of 2^32? (a) 6 (b) 8 (c) 2 (d) 4
19. 19.What is the smallest number you must add to 116 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
20. 20.What is the remainder when 258 is divided by 12? (a) 8 (b) 5 (c) 6 (d) 7
21. 21.What is the units digit of 3^60? (a) 7 (b) 9 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 11? (a) 9690 (b) 9700 (c) 9680 (d) 9675
23. 23.What is the smallest number you must add to 354 to make it divisible by 7? (a) 4 (b) 3 (c) 2 (d) 5
24. 24.What is the units digit of 4^21? (a) 5 (b) 6 (c) 3 (d) 4
25. 25.What is the units digit of 9^17? (a) 7 (b) 9 (c) 1 (d) 8
26. 26.Which of these numbers is divisible by 9? (a) 1968 (b) 1985 (c) 1979 (d) 1971
27. 27.Which of these numbers is divisible by 3? (a) 317 (b) 315 (c) 319 (d) 314
28. 28.What is the smallest number you must add to 364 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
29. 29.What is the remainder when 494 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1
30. 30.What is the units digit of 2^10? (a) 4 (b) 8 (c) 6 (d) 2
31. 31.What is the units digit of 8^15? (a) 8 (b) 6 (c) 2 (d) 4
32. 32.What is the smallest number you must add to 715 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
33. 33.What is the units digit of 3^27? (a) 3 (b) 1 (c) 7 (d) 9
34. 34.What is the smallest number you must add to 315 to make it divisible by 8? (a) 6 (b) 3 (c) 5 (d) 4
35. 35.Which of these numbers is divisible by 3? (a) 293 (b) 290 (c) 286 (d) 288
36. 36.What is the smallest number you must add to 150 to make it divisible by 8? (a) 6 (b) 3 (c) 1 (d) 2
37. 37.What is the remainder when 172 is divided by 3? (a) 1 (b) 0 (c) 2
38. 38.Which of these numbers is divisible by 4? (a) 540 (b) 547 (c) 541 (d) 537
39. 39.What is the smallest number you must add to 463 to make it divisible by 12? (a) 4 (b) 5 (c) 6 (d) 7
40. 40.Which of these numbers is divisible by 6? (a) 1081 (b) 1089 (c) 1093 (d) 1086

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