Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 418 · 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 312 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
2. 2.Which of these numbers is divisible by 3? (a) 241 (b) 237 (c) 239 (d) 235
3. 3.What is the smallest number you must add to 157 to make it divisible by 11? (a) 8 (b) 9 (c) 3 (d) 7
4. 4.Which of these numbers is divisible by 6? (a) 4118 (b) 4122 (c) 4126 (d) 4133
5. 5.What is the remainder when 687 is divided by 10? (a) 6 (b) 7 (c) 3 (d) 8
6. 6.Which of these numbers is divisible by 6? (a) 2781 (b) 2778 (c) 2785 (d) 2773
7. 7.What is the units digit of 7^54? (a) 1 (b) 9 (c) 3 (d) 7
8. 8.Which of these numbers is divisible by 11? (a) 1540 (b) 1548 (c) 1558 (d) 1530
9. 9.What is the remainder when 629 is divided by 8? (a) 4 (b) 5 (c) 6 (d) 3
10. 10.What is the units digit of 4^44? (a) 6 (b) 5 (c) 7 (d) 4
11. 11.Which of these numbers is divisible by 3? (a) 805 (b) 812 (c) 809 (d) 807
12. 12.What is the smallest number you must add to 211 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the units digit of 2^50? (a) 4 (b) 8 (c) 2 (d) 6
14. 14.Which of these numbers is divisible by 3? (a) 457 (b) 454 (c) 453 (d) 451
15. 15.What is the remainder when 565 is divided by 10? (a) 5 (b) 7 (c) 4 (d) 6
16. 16.What is the smallest number you must add to 341 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
17. 17.What is the remainder when 631 is divided by 11? (a) 3 (b) 5 (c) 4 (d) 7
18. 18.What is the units digit of 8^58? (a) 6 (b) 4 (c) 8 (d) 2
19. 19.What is the units digit of 2^54? (a) 8 (b) 4 (c) 6 (d) 2
20. 20.What is the units digit of 7^58? (a) 3 (b) 1 (c) 7 (d) 9
21. 21.What is the smallest number you must add to 558 to make it divisible by 8? (a) 3 (b) 6 (c) 2 (d) 1
22. 22.Which of these numbers is divisible by 4? (a) 1493 (b) 1492 (c) 1489 (d) 1498
23. 23.Which of these numbers is divisible by 3? (a) 209 (b) 212 (c) 206 (d) 207
24. 24.Which of these numbers is divisible by 3? (a) 1382 (b) 1380 (c) 1384 (d) 1378
25. 25.What is the units digit of 2^13? (a) 4 (b) 8 (c) 2 (d) 6
26. 26.What is the units digit of 4^29? (a) 5 (b) 4 (c) 3 (d) 6
27. 27.Which of these numbers is divisible by 6? (a) 4819 (b) 4824 (c) 4829 (d) 4832
28. 28.What is the remainder when 185 is divided by 10? (a) 4 (b) 7 (c) 6 (d) 5
29. 29.What is the units digit of 9^56? (a) 0 (b) 1 (c) 2 (d) 9
30. 30.What is the remainder when 83 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
31. 31.What is the smallest number you must add to 369 to make it divisible by 12? (a) 2 (b) 4 (c) 3 (d) 9
32. 32.What is the remainder when 773 is divided by 10? (a) 3 (b) 4 (c) 2 (d) 7
33. 33.What is the remainder when 77 is divided by 3? (a) 1 (b) 2 (c) 0
34. 34.Which of these numbers is divisible by 4? (a) 3335 (b) 3328 (c) 3329 (d) 3326
35. 35.Which of these numbers is divisible by 4? (a) 2170 (b) 2174 (c) 2178 (d) 2172
36. 36.What is the smallest number you must add to 59 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
37. 37.Which of these numbers is divisible by 8? (a) 4534 (b) 4549 (c) 4543 (d) 4536
38. 38.What is the remainder when 162 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
39. 39.What is the remainder when 278 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
40. 40.Which of these numbers is divisible by 11? (a) 3524 (b) 3516 (c) 3506 (d) 3509

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