Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 350 · 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 11? (a) 5977 (b) 5993 (c) 5968 (d) 5973
2. 2.What is the remainder when 306 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
3. 3.Which of these numbers is divisible by 8? (a) 2168 (b) 2167 (c) 2169 (d) 2177
4. 4.Which of these numbers is divisible by 11? (a) 457 (b) 449 (c) 440 (d) 432
5. 5.Which of these numbers is divisible by 4? (a) 3189 (b) 3192 (c) 3194 (d) 3198
6. 6.Which of these numbers is divisible by 3? (a) 1887 (b) 1888 (c) 1885 (d) 1891
7. 7.Which of these numbers is divisible by 6? (a) 1544 (b) 1531 (c) 1540 (d) 1536
8. 8.What is the units digit of 2^10? (a) 2 (b) 4 (c) 8 (d) 6
9. 9.What is the smallest number you must add to 477 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
10. 10.What is the remainder when 685 is divided by 10? (a) 7 (b) 6 (c) 4 (d) 5
11. 11.What is the remainder when 458 is divided by 9? (a) 8 (b) 6 (c) 7 (d) 1
12. 12.What is the units digit of 2^11? (a) 4 (b) 6 (c) 8 (d) 2
13. 13.What is the smallest number you must add to 389 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 4
14. 14.Which of these numbers is divisible by 3? (a) 1838 (b) 1831 (c) 1833 (d) 1835
15. 15.What is the units digit of 9^17? (a) 1 (b) 7 (c) 9 (d) 8
16. 16.What is the remainder when 794 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
17. 17.Which of these numbers is divisible by 9? (a) 455 (b) 463 (c) 444 (d) 450
18. 18.What is the remainder when 332 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
19. 19.What is the smallest number you must add to 441 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
20. 20.Which of these numbers is divisible by 8? (a) 1612 (b) 1595 (c) 1600 (d) 1605
21. 21.What is the smallest number you must add to 401 to make it divisible by 10? (a) 1 (b) 7 (c) 9 (d) 8
22. 22.Which of these numbers is divisible by 4? (a) 829 (b) 828 (c) 825 (d) 833
23. 23.What is the smallest number you must add to 272 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 2
24. 24.What is the remainder when 393 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
25. 25.What is the units digit of 8^23? (a) 2 (b) 8 (c) 4 (d) 6
26. 26.What is the units digit of 8^31? (a) 8 (b) 2 (c) 4 (d) 6
27. 27.What is the smallest number you must add to 408 to make it divisible by 7? (a) 5 (b) 2 (c) 6 (d) 4
28. 28.What is the smallest number you must add to 358 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
29. 29.What is the units digit of 7^39? (a) 1 (b) 9 (c) 3 (d) 7
30. 30.What is the smallest number you must add to 569 to make it divisible by 11? (a) 8 (b) 4 (c) 3 (d) 2
31. 31.Which of these numbers is divisible by 3? (a) 1190 (b) 1186 (c) 1188 (d) 1192
32. 32.What is the smallest number you must add to 521 to make it divisible by 7? (a) 3 (b) 6 (c) 5 (d) 4
33. 33.What is the units digit of 7^37? (a) 7 (b) 9 (c) 1 (d) 3
34. 34.What is the remainder when 291 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
35. 35.What is the remainder when 210 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
36. 36.What is the remainder when 127 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
37. 37.What is the remainder when 149 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
38. 38.Which of these numbers is divisible by 4? (a) 1297 (b) 1296 (c) 1301 (d) 1294
39. 39.What is the smallest number you must add to 133 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
40. 40.Which of these numbers is divisible by 3? (a) 1405 (b) 1407 (c) 1408 (d) 1411

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