Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 475 · 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 137 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
2. 2.What is the smallest number you must add to 198 to make it divisible by 4? (a) 3 (b) 2 (c) 1
3. 3.What is the smallest number you must add to 128 to make it divisible by 12? (a) 4 (b) 8 (c) 5 (d) 3
4. 4.What is the remainder when 221 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
5. 5.What is the units digit of 9^52? (a) 2 (b) 1 (c) 0 (d) 9
6. 6.Which of these numbers is divisible by 8? (a) 4117 (b) 4120 (c) 4133 (d) 4122
7. 7.What is the smallest number you must add to 344 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
8. 8.Which of these numbers is divisible by 3? (a) 1752 (b) 1757 (c) 1751 (d) 1753
9. 9.What is the remainder when 125 is divided by 3? (a) 1 (b) 0 (c) 2
10. 10.Which of these numbers is divisible by 6? (a) 3551 (b) 3552 (c) 3559 (d) 3557
11. 11.Which of these numbers is divisible by 4? (a) 2530 (b) 2528 (c) 2525 (d) 2534
12. 12.Which of these numbers is divisible by 8? (a) 374 (b) 360 (c) 357 (d) 361
13. 13.What is the units digit of 7^10? (a) 9 (b) 1 (c) 3 (d) 7
14. 14.What is the smallest number you must add to 202 to make it divisible by 11? (a) 4 (b) 6 (c) 7 (d) 8
15. 15.What is the remainder when 188 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
16. 16.Which of these numbers is divisible by 9? (a) 442 (b) 441 (c) 433 (d) 455
17. 17.What is the units digit of 4^54? (a) 4 (b) 6 (c) 7 (d) 5
18. 18.What is the units digit of 2^30? (a) 8 (b) 6 (c) 4 (d) 2
19. 19.What is the remainder when 641 is divided by 9? (a) 2 (b) 1 (c) 7 (d) 3
20. 20.Which of these numbers is divisible by 8? (a) 6123 (b) 6116 (c) 6129 (d) 6120
21. 21.What is the remainder when 314 is divided by 11? (a) 6 (b) 5 (c) 7 (d) 8
22. 22.What is the units digit of 8^10? (a) 8 (b) 4 (c) 6 (d) 2
23. 23.What is the remainder when 435 is divided by 9? (a) 3 (b) 2 (c) 6 (d) 4
24. 24.What is the units digit of 2^28? (a) 4 (b) 6 (c) 8 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 1472 (b) 1468 (c) 1475 (d) 1470
26. 26.Which of these numbers is divisible by 4? (a) 1885 (b) 1882 (c) 1879 (d) 1880
27. 27.Which of these numbers is divisible by 6? (a) 4592 (b) 4585 (c) 4584 (d) 4582
28. 28.What is the smallest number you must add to 194 to make it divisible by 3? (a) 2 (b) 1
29. 29.What is the smallest number you must add to 291 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
30. 30.What is the units digit of 9^55? (a) 8 (b) 1 (c) 9 (d) 7
31. 31.What is the remainder when 471 is divided by 11? (a) 2 (b) 10 (c) 8 (d) 9
32. 32.What is the remainder when 235 is divided by 3? (a) 2 (b) 0 (c) 1
33. 33.What is the units digit of 4^14? (a) 4 (b) 6 (c) 5 (d) 7
34. 34.What is the smallest number you must add to 93 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
35. 35.Which of these numbers is divisible by 8? (a) 2125 (b) 2120 (c) 2129 (d) 2113
36. 36.What is the smallest number you must add to 211 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
37. 37.What is the units digit of 8^23? (a) 2 (b) 6 (c) 8 (d) 4
38. 38.Which of these numbers is divisible by 3? (a) 258 (b) 260 (c) 262 (d) 257
39. 39.What is the smallest number you must add to 316 to make it divisible by 11? (a) 8 (b) 3 (c) 2 (d) 4
40. 40.Which of these numbers is divisible by 8? (a) 5227 (b) 5211 (c) 5218 (d) 5216

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