Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 64 · 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) 386 (b) 387 (c) 389 (d) 391
2. 2.Which of these numbers is divisible by 4? (a) 2157 (b) 2151 (c) 2154 (d) 2152
3. 3.What is the smallest number you must add to 419 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
4. 4.What is the remainder when 259 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
5. 5.What is the remainder when 496 is divided by 9? (a) 1 (b) 0 (c) 2 (d) 8
6. 6.Which of these numbers is divisible by 6? (a) 5136 (b) 5140 (c) 5143 (d) 5132
7. 7.What is the remainder when 217 is divided by 8? (a) 2 (b) 1 (c) 7 (d) 0
8. 8.What is the units digit of 2^37? (a) 6 (b) 2 (c) 4 (d) 8
9. 9.Which of these numbers is divisible by 11? (a) 986 (b) 972 (c) 968 (d) 967
10. 10.What is the remainder when 262 is divided by 9? (a) 0 (b) 8 (c) 1 (d) 2
11. 11.What is the smallest number you must add to 333 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
12. 12.What is the remainder when 638 is divided by 8? (a) 5 (b) 2 (c) 6 (d) 7
13. 13.What is the smallest number you must add to 286 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
14. 14.Which of these numbers is divisible by 4? (a) 1087 (b) 1079 (c) 1080 (d) 1082
15. 15.Which of these numbers is divisible by 4? (a) 2535 (b) 2537 (c) 2532 (d) 2529
16. 16.Which of these numbers is divisible by 4? (a) 291 (b) 287 (c) 284 (d) 282
17. 17.What is the units digit of 2^57? (a) 8 (b) 2 (c) 4 (d) 6
18. 18.What is the remainder when 554 is divided by 12? (a) 10 (b) 2 (c) 1 (d) 3
19. 19.What is the smallest number you must add to 82 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
20. 20.Which of these numbers is divisible by 3? (a) 1214 (b) 1207 (c) 1209 (d) 1211
21. 21.Which of these numbers is divisible by 9? (a) 6662 (b) 6677 (c) 6660 (d) 6653
22. 22.What is the smallest number you must add to 648 to make it divisible by 10? (a) 2 (b) 1 (c) 3 (d) 8
23. 23.What is the remainder when 147 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
24. 24.Which of these numbers is divisible by 11? (a) 3290 (b) 3289 (c) 3307 (d) 3283
25. 25.Which of these numbers is divisible by 8? (a) 2635 (b) 2625 (c) 2624 (d) 2621
26. 26.What is the units digit of 8^49? (a) 8 (b) 6 (c) 4 (d) 2
27. 27.Which of these numbers is divisible by 8? (a) 2689 (b) 2698 (c) 2696 (d) 2709
28. 28.Which of these numbers is divisible by 6? (a) 2118 (b) 2120 (c) 2126 (d) 2116
29. 29.What is the remainder when 82 is divided by 3? (a) 2 (b) 1 (c) 0
30. 30.What is the smallest number you must add to 126 to make it divisible by 4? (a) 2 (b) 1 (c) 3
31. 31.What is the smallest number you must add to 647 to make it divisible by 8? (a) 7 (b) 3 (c) 2 (d) 1
32. 32.What is the units digit of 2^15? (a) 8 (b) 4 (c) 2 (d) 6
33. 33.What is the units digit of 8^17? (a) 8 (b) 4 (c) 2 (d) 6
34. 34.What is the remainder when 614 is divided by 12? (a) 3 (b) 1 (c) 10 (d) 2
35. 35.What is the remainder when 93 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
36. 36.What is the remainder when 112 is divided by 3? (a) 1 (b) 2 (c) 0
37. 37.What is the units digit of 4^58? (a) 6 (b) 4 (c) 7 (d) 5
38. 38.What is the remainder when 284 is divided by 6? (a) 2 (b) 1 (c) 4 (d) 3
39. 39.Which of these numbers is divisible by 11? (a) 4868 (b) 4891 (c) 4873 (d) 4874
40. 40.What is the remainder when 739 is divided by 11? (a) 1 (b) 3 (c) 2 (d) 9

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