Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 979 · 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) 268 (b) 275 (c) 270 (d) 271
2. 2.What is the remainder when 236 is divided by 5? (a) 1 (b) 4 (c) 0 (d) 2
3. 3.Which of these numbers is divisible by 3? (a) 597 (b) 599 (c) 596 (d) 602
4. 4.Which of these numbers is divisible by 8? (a) 4681 (b) 4670 (c) 4674 (d) 4672
5. 5.What is the units digit of 4^23? (a) 3 (b) 5 (c) 6 (d) 4
6. 6.Which of these numbers is divisible by 11? (a) 7876 (b) 7882 (c) 7888 (d) 7866
7. 7.What is the remainder when 530 is divided by 9? (a) 1 (b) 6 (c) 7 (d) 8
8. 8.What is the smallest number you must add to 179 to make it divisible by 4? (a) 3 (b) 2 (c) 1
9. 9.What is the units digit of 7^31? (a) 7 (b) 1 (c) 3 (d) 9
10. 10.What is the smallest number you must add to 497 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
11. 11.What is the remainder when 166 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
12. 12.Which of these numbers is divisible by 11? (a) 8029 (b) 8033 (c) 8042 (d) 8030
13. 13.What is the smallest number you must add to 226 to make it divisible by 3? (a) 1 (b) 2
14. 14.What is the units digit of 3^38? (a) 7 (b) 1 (c) 9 (d) 3
15. 15.What is the remainder when 257 is divided by 7? (a) 2 (b) 4 (c) 6 (d) 5
16. 16.What is the smallest number you must add to 495 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 7
17. 17.What is the remainder when 128 is divided by 10? (a) 9 (b) 2 (c) 7 (d) 8
18. 18.What is the remainder when 181 is divided by 3? (a) 0 (b) 2 (c) 1
19. 19.Which of these numbers is divisible by 6? (a) 1252 (b) 1262 (c) 1256 (d) 1254
20. 20.What is the remainder when 134 is divided by 12? (a) 10 (b) 3 (c) 2 (d) 1
21. 21.Which of these numbers is divisible by 6? (a) 267 (b) 271 (c) 279 (d) 270
22. 22.What is the units digit of 8^57? (a) 6 (b) 8 (c) 2 (d) 4
23. 23.Which of these numbers is divisible by 11? (a) 3523 (b) 3520 (c) 3532 (d) 3519
24. 24.What is the units digit of 4^51? (a) 6 (b) 3 (c) 5 (d) 4
25. 25.What is the smallest number you must add to 682 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 7
26. 26.What is the remainder when 67 is divided by 3? (a) 0 (b) 2 (c) 1
27. 27.Which of these numbers is divisible by 8? (a) 2359 (b) 2360 (c) 2374 (d) 2365
28. 28.What is the smallest number you must add to 843 to make it divisible by 11? (a) 3 (b) 4 (c) 5 (d) 7
29. 29.Which of these numbers is divisible by 4? (a) 1731 (b) 1727 (c) 1728 (d) 1733
30. 30.Which of these numbers is divisible by 9? (a) 2583 (b) 2580 (c) 2593 (d) 2591
31. 31.What is the smallest number you must add to 239 to make it divisible by 3? (a) 2 (b) 1
32. 32.Which of these numbers is divisible by 4? (a) 260 (b) 261 (c) 266 (d) 257
33. 33.What is the units digit of 9^45? (a) 1 (b) 9 (c) 7 (d) 8
34. 34.What is the smallest number you must add to 149 to make it divisible by 3? (a) 1 (b) 2
35. 35.What is the remainder when 781 is divided by 12? (a) 1 (b) 0 (c) 11 (d) 2
36. 36.What is the remainder when 573 is divided by 10? (a) 4 (b) 3 (c) 7 (d) 2
37. 37.What is the remainder when 562 is divided by 11? (a) 10 (b) 2 (c) 0 (d) 1
38. 38.What is the units digit of 3^34? (a) 7 (b) 3 (c) 9 (d) 1
39. 39.What is the remainder when 709 is divided by 11? (a) 5 (b) 6 (c) 4 (d) 7
40. 40.What is the units digit of 7^24? (a) 1 (b) 9 (c) 7 (d) 3

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