Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 126 · 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 smallest number you must add to 251 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 1
2. 2.What is the remainder when 618 is divided by 10? (a) 7 (b) 2 (c) 8 (d) 9
3. 3.Which of these numbers is divisible by 11? (a) 608 (b) 604 (c) 594 (d) 586
4. 4.What is the units digit of 9^12? (a) 9 (b) 2 (c) 1 (d) 0
5. 5.Which of these numbers is divisible by 4? (a) 3143 (b) 3147 (c) 3140 (d) 3137
6. 6.What is the remainder when 378 is divided by 10? (a) 7 (b) 2 (c) 9 (d) 8
7. 7.What is the remainder when 276 is divided by 11? (a) 2 (b) 0 (c) 1 (d) 10
8. 8.Which of these numbers is divisible by 8? (a) 2896 (b) 2905 (c) 2898 (d) 2893
9. 9.What is the smallest number you must add to 230 to make it divisible by 4? (a) 1 (b) 3 (c) 2
10. 10.What is the units digit of 8^40? (a) 4 (b) 8 (c) 2 (d) 6
11. 11.Which of these numbers is divisible by 4? (a) 739 (b) 743 (c) 745 (d) 740
12. 12.What is the smallest number you must add to 145 to make it divisible by 3? (a) 1 (b) 2
13. 13.Which of these numbers is divisible by 11? (a) 6577 (b) 6584 (c) 6595 (d) 6578
14. 14.What is the smallest number you must add to 149 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 6
15. 15.What is the smallest number you must add to 268 to make it divisible by 12? (a) 8 (b) 9 (c) 7 (d) 4
16. 16.What is the remainder when 309 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
17. 17.What is the smallest number you must add to 293 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
18. 18.Which of these numbers is divisible by 8? (a) 4998 (b) 4985 (c) 5003 (d) 4992
19. 19.What is the remainder when 637 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
20. 20.Which of these numbers is divisible by 3? (a) 2016 (b) 2018 (c) 2015 (d) 2020
21. 21.Which of these numbers is divisible by 11? (a) 9539 (b) 9537 (c) 9531 (d) 9554
22. 22.Which of these numbers is divisible by 6? (a) 2362 (b) 2356 (c) 2349 (d) 2352
23. 23.Which of these numbers is divisible by 3? (a) 344 (b) 345 (c) 349 (d) 347
24. 24.Which of these numbers is divisible by 11? (a) 713 (b) 721 (c) 729 (d) 715
25. 25.What is the units digit of 8^12? (a) 4 (b) 6 (c) 2 (d) 8
26. 26.What is the units digit of 2^45? (a) 6 (b) 2 (c) 4 (d) 8
27. 27.What is the units digit of 7^37? (a) 7 (b) 1 (c) 3 (d) 9
28. 28.What is the units digit of 4^11? (a) 6 (b) 3 (c) 4 (d) 5
29. 29.Which of these numbers is divisible by 4? (a) 3297 (b) 3300 (c) 3301 (d) 3305
30. 30.What is the units digit of 3^59? (a) 9 (b) 3 (c) 1 (d) 7
31. 31.What is the smallest number you must add to 398 to make it divisible by 8? (a) 6 (b) 2 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 6? (a) 3081 (b) 3092 (c) 3089 (d) 3084
33. 33.What is the remainder when 933 is divided by 12? (a) 10 (b) 8 (c) 3 (d) 9
34. 34.What is the smallest number you must add to 573 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 1
35. 35.What is the remainder when 253 is divided by 12? (a) 2 (b) 1 (c) 0 (d) 11
36. 36.What is the remainder when 324 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
37. 37.What is the units digit of 2^21? (a) 6 (b) 4 (c) 2 (d) 8
38. 38.What is the smallest number you must add to 677 to make it divisible by 10? (a) 7 (b) 3 (c) 4 (d) 2
39. 39.What is the units digit of 4^10? (a) 6 (b) 5 (c) 4 (d) 7
40. 40.What is the remainder when 451 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1

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