Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 201 · 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) 735 (b) 733 (c) 736 (d) 739
2. 2.What is the remainder when 85 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
3. 3.Which of these numbers is divisible by 8? (a) 2586 (b) 2607 (c) 2592 (d) 2593
4. 4.Which of these numbers is divisible by 3? (a) 2565 (b) 2566 (c) 2569 (d) 2564
5. 5.What is the units digit of 8^33? (a) 8 (b) 2 (c) 6 (d) 4
6. 6.What is the smallest number you must add to 166 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.What is the remainder when 59 is divided by 3? (a) 1 (b) 0 (c) 2
8. 8.Which of these numbers is divisible by 4? (a) 2744 (b) 2741 (c) 2746 (d) 2751
9. 9.Which of these numbers is divisible by 11? (a) 5775 (b) 5770 (c) 5790 (d) 5776
10. 10.Which of these numbers is divisible by 6? (a) 4896 (b) 4907 (c) 4899 (d) 4895
11. 11.What is the smallest number you must add to 699 to make it divisible by 9? (a) 6 (b) 4 (c) 2 (d) 3
12. 12.What is the smallest number you must add to 249 to make it divisible by 4? (a) 2 (b) 3 (c) 1
13. 13.What is the units digit of 8^37? (a) 2 (b) 4 (c) 6 (d) 8
14. 14.What is the remainder when 165 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
15. 15.What is the remainder when 182 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
16. 16.What is the units digit of 7^20? (a) 9 (b) 3 (c) 1 (d) 7
17. 17.What is the remainder when 332 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
18. 18.What is the remainder when 181 is divided by 3? (a) 0 (b) 1 (c) 2
19. 19.Which of these numbers is divisible by 8? (a) 6454 (b) 6461 (c) 6456 (d) 6467
20. 20.What is the units digit of 4^10? (a) 5 (b) 7 (c) 4 (d) 6
21. 21.What is the smallest number you must add to 255 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 7
22. 22.Which of these numbers is divisible by 6? (a) 3025 (b) 3021 (c) 3031 (d) 3024
23. 23.What is the remainder when 515 is divided by 12? (a) 9 (b) 1 (c) 11 (d) 10
24. 24.What is the remainder when 181 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
25. 25.What is the units digit of 2^12? (a) 2 (b) 8 (c) 4 (d) 6
26. 26.What is the smallest number you must add to 130 to make it divisible by 11? (a) 1 (b) 3 (c) 2 (d) 9
27. 27.What is the units digit of 4^23? (a) 5 (b) 3 (c) 6 (d) 4
28. 28.Which of these numbers is divisible by 4? (a) 437 (b) 432 (c) 431 (d) 434
29. 29.Which of these numbers is divisible by 4? (a) 2741 (b) 2746 (c) 2737 (d) 2740
30. 30.What is the remainder when 134 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
31. 31.Which of these numbers is divisible by 11? (a) 1981 (b) 1973 (c) 1999 (d) 1980
32. 32.Which of these numbers is divisible by 3? (a) 425 (b) 427 (c) 426 (d) 431
33. 33.What is the smallest number you must add to 400 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
34. 34.Which of these numbers is divisible by 11? (a) 1413 (b) 1421 (c) 1404 (d) 1408
35. 35.What is the smallest number you must add to 236 to make it divisible by 11? (a) 5 (b) 6 (c) 7 (d) 8
36. 36.What is the smallest number you must add to 364 to make it divisible by 10? (a) 5 (b) 4 (c) 7 (d) 6
37. 37.What is the smallest number you must add to 714 to make it divisible by 9? (a) 6 (b) 7 (c) 5 (d) 3
38. 38.What is the units digit of 2^11? (a) 8 (b) 6 (c) 2 (d) 4
39. 39.What is the smallest number you must add to 655 to make it divisible by 12? (a) 6 (b) 5 (c) 4 (d) 7
40. 40.What is the smallest number you must add to 396 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 3

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