Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 57 · 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 8? (a) 3235 (b) 3232 (c) 3241 (d) 3231
2. 2.Which of these numbers is divisible by 9? (a) 5400 (b) 5395 (c) 5411 (d) 5404
3. 3.What is the remainder when 346 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 0
4. 4.What is the remainder when 681 is divided by 11? (a) 1 (b) 8 (c) 9 (d) 10
5. 5.What is the smallest number you must add to 322 to make it divisible by 4? (a) 3 (b) 1 (c) 2
6. 6.What is the units digit of 8^60? (a) 2 (b) 8 (c) 4 (d) 6
7. 7.What is the remainder when 242 is divided by 12? (a) 3 (b) 2 (c) 1 (d) 10
8. 8.What is the smallest number you must add to 146 to make it divisible by 3? (a) 2 (b) 1
9. 9.Which of these numbers is divisible by 6? (a) 3379 (b) 3371 (c) 3376 (d) 3372
10. 10.Which of these numbers is divisible by 6? (a) 3287 (b) 3296 (c) 3288 (d) 3291
11. 11.What is the remainder when 209 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
12. 12.What is the remainder when 164 is divided by 3? (a) 0 (b) 1 (c) 2
13. 13.What is the smallest number you must add to 864 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 5
14. 14.What is the units digit of 4^34? (a) 7 (b) 5 (c) 6 (d) 4
15. 15.Which of these numbers is divisible by 11? (a) 7180 (b) 7172 (c) 7192 (d) 7166
16. 16.What is the units digit of 2^10? (a) 6 (b) 2 (c) 4 (d) 8
17. 17.Which of these numbers is divisible by 8? (a) 4234 (b) 4224 (c) 4223 (d) 4227
18. 18.What is the units digit of 8^39? (a) 2 (b) 4 (c) 6 (d) 8
19. 19.Which of these numbers is divisible by 4? (a) 2081 (b) 2079 (c) 2086 (d) 2080
20. 20.What is the units digit of 8^50? (a) 8 (b) 6 (c) 4 (d) 2
21. 21.What is the remainder when 282 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
22. 22.What is the units digit of 3^51? (a) 7 (b) 9 (c) 1 (d) 3
23. 23.What is the remainder when 715 is divided by 10? (a) 5 (b) 6 (c) 4 (d) 7
24. 24.Which of these numbers is divisible by 11? (a) 3498 (b) 3514 (c) 3503 (d) 3491
25. 25.What is the remainder when 285 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
26. 26.What is the remainder when 77 is divided by 3? (a) 0 (b) 2 (c) 1
27. 27.Which of these numbers is divisible by 8? (a) 1830 (b) 1824 (c) 1823 (d) 1837
28. 28.Which of these numbers is divisible by 6? (a) 4590 (b) 4587 (c) 4594 (d) 4599
29. 29.What is the remainder when 73 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
30. 30.Which of these numbers is divisible by 3? (a) 2526 (b) 2530 (c) 2524 (d) 2527
31. 31.What is the units digit of 8^18? (a) 8 (b) 4 (c) 2 (d) 6
32. 32.What is the units digit of 7^16? (a) 3 (b) 7 (c) 9 (d) 1
33. 33.What is the remainder when 38 is divided by 3? (a) 2 (b) 0 (c) 1
34. 34.What is the units digit of 3^49? (a) 1 (b) 3 (c) 9 (d) 7
35. 35.What is the remainder when 470 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
36. 36.Which of these numbers is divisible by 8? (a) 4056 (b) 4060 (c) 4066 (d) 4053
37. 37.What is the units digit of 7^54? (a) 9 (b) 1 (c) 7 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 2296 (b) 2298 (c) 2293 (d) 2302
39. 39.What is the remainder when 103 is divided by 6? (a) 0 (b) 1 (c) 2 (d) 5
40. 40.What is the smallest number you must add to 726 to make it divisible by 9? (a) 4 (b) 6 (c) 2 (d) 3

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