Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 160 · 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 162 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
2. 2.What is the smallest number you must add to 466 to make it divisible by 9? (a) 3 (b) 1 (c) 7 (d) 2
3. 3.What is the units digit of 2^23? (a) 8 (b) 4 (c) 2 (d) 6
4. 4.What is the smallest number you must add to 106 to make it divisible by 9? (a) 2 (b) 1 (c) 7 (d) 3
5. 5.What is the units digit of 4^50? (a) 6 (b) 4 (c) 7 (d) 5
6. 6.Which of these numbers is divisible by 11? (a) 7382 (b) 7367 (c) 7374 (d) 7370
7. 7.What is the remainder when 83 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 2
8. 8.What is the remainder when 104 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 1
9. 9.Which of these numbers is divisible by 8? (a) 1129 (b) 1136 (c) 1141 (d) 1151
10. 10.Which of these numbers is divisible by 8? (a) 2519 (b) 2509 (c) 2501 (d) 2504
11. 11.What is the units digit of 4^13? (a) 3 (b) 6 (c) 4 (d) 5
12. 12.What is the units digit of 9^24? (a) 1 (b) 0 (c) 2 (d) 9
13. 13.What is the remainder when 263 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 287 (b) 284 (c) 282 (d) 281
15. 15.Which of these numbers is divisible by 4? (a) 2368 (b) 2365 (c) 2371 (d) 2373
16. 16.Which of these numbers is divisible by 8? (a) 6218 (b) 6216 (c) 6226 (d) 6209
17. 17.What is the remainder when 122 is divided by 8? (a) 1 (b) 3 (c) 2 (d) 6
18. 18.What is the units digit of 3^16? (a) 1 (b) 9 (c) 3 (d) 7
19. 19.What is the smallest number you must add to 129 to make it divisible by 10? (a) 1 (b) 2 (c) 9 (d) 3
20. 20.What is the units digit of 2^53? (a) 4 (b) 2 (c) 8 (d) 6
21. 21.What is the remainder when 285 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
22. 22.What is the units digit of 3^49? (a) 9 (b) 7 (c) 3 (d) 1
23. 23.Which of these numbers is divisible by 3? (a) 662 (b) 665 (c) 660 (d) 658
24. 24.What is the remainder when 125 is divided by 9? (a) 7 (b) 6 (c) 1 (d) 8
25. 25.Which of these numbers is divisible by 8? (a) 3209 (b) 3205 (c) 3218 (d) 3208
26. 26.What is the units digit of 7^45? (a) 1 (b) 3 (c) 9 (d) 7
27. 27.What is the smallest number you must add to 50 to make it divisible by 3? (a) 2 (b) 1
28. 28.Which of these numbers is divisible by 9? (a) 1644 (b) 1655 (c) 1632 (d) 1638
29. 29.What is the units digit of 7^50? (a) 1 (b) 9 (c) 7 (d) 3
30. 30.What is the units digit of 3^57? (a) 1 (b) 9 (c) 7 (d) 3
31. 31.What is the remainder when 80 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
32. 32.What is the remainder when 299 is divided by 12? (a) 10 (b) 11 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 6? (a) 3373 (b) 3363 (c) 3366 (d) 3368
34. 34.What is the units digit of 7^19? (a) 9 (b) 1 (c) 7 (d) 3
35. 35.Which of these numbers is divisible by 11? (a) 1899 (b) 1873 (c) 1886 (d) 1881
36. 36.What is the remainder when 460 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
37. 37.What is the remainder when 328 is divided by 10? (a) 2 (b) 8 (c) 9 (d) 7
38. 38.What is the smallest number you must add to 207 to make it divisible by 4? (a) 2 (b) 1 (c) 3
39. 39.What is the units digit of 9^45? (a) 7 (b) 1 (c) 9 (d) 8
40. 40.What is the smallest number you must add to 502 to make it divisible by 10? (a) 8 (b) 9 (c) 7 (d) 2

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