Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 791 · 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 119 is divided by 10? (a) 7 (b) 1 (c) 9 (d) 8
2. 2.Which of these numbers is divisible by 6? (a) 3522 (b) 3530 (c) 3518 (d) 3525
3. 3.What is the smallest number you must add to 47 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the remainder when 718 is divided by 9? (a) 2 (b) 7 (c) 8 (d) 6
5. 5.What is the remainder when 719 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 1
6. 6.What is the remainder when 106 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
7. 7.What is the units digit of 4^21? (a) 4 (b) 6 (c) 3 (d) 5
8. 8.Which of these numbers is divisible by 4? (a) 1163 (b) 1169 (c) 1166 (d) 1164
9. 9.What is the units digit of 8^58? (a) 4 (b) 8 (c) 2 (d) 6
10. 10.What is the units digit of 7^10? (a) 3 (b) 1 (c) 7 (d) 9
11. 11.What is the remainder when 106 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
12. 12.What is the smallest number you must add to 423 to make it divisible by 11? (a) 5 (b) 6 (c) 8 (d) 7
13. 13.What is the smallest number you must add to 663 to make it divisible by 11? (a) 3 (b) 9 (c) 8 (d) 7
14. 14.What is the units digit of 9^17? (a) 9 (b) 8 (c) 1 (d) 7
15. 15.Which of these numbers is divisible by 3? (a) 1048 (b) 1051 (c) 1045 (d) 1047
16. 16.Which of these numbers is divisible by 3? (a) 1781 (b) 1777 (c) 1784 (d) 1779
17. 17.What is the remainder when 77 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
18. 18.Which of these numbers is divisible by 6? (a) 2089 (b) 2095 (c) 2094 (d) 2105
19. 19.What is the smallest number you must add to 101 to make it divisible by 3? (a) 1 (b) 2
20. 20.What is the smallest number you must add to 681 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 1
21. 21.Which of these numbers is divisible by 6? (a) 2105 (b) 2095 (c) 2110 (d) 2100
22. 22.What is the remainder when 75 is divided by 7? (a) 5 (b) 2 (c) 6 (d) 4
23. 23.What is the smallest number you must add to 301 to make it divisible by 10? (a) 1 (b) 7 (c) 9 (d) 8
24. 24.What is the remainder when 938 is divided by 12? (a) 2 (b) 1 (c) 10 (d) 3
25. 25.What is the smallest number you must add to 777 to make it divisible by 10? (a) 2 (b) 4 (c) 3 (d) 7
26. 26.Which of these numbers is divisible by 6? (a) 645 (b) 653 (c) 648 (d) 655
27. 27.Which of these numbers is divisible by 9? (a) 778 (b) 787 (c) 774 (d) 769
28. 28.What is the units digit of 9^14? (a) 1 (b) 2 (c) 0 (d) 9
29. 29.Which of these numbers is divisible by 3? (a) 1822 (b) 1829 (c) 1825 (d) 1824
30. 30.Which of these numbers is divisible by 4? (a) 1869 (b) 1868 (c) 1873 (d) 1865
31. 31.What is the smallest number you must add to 341 to make it divisible by 7? (a) 5 (b) 3 (c) 1 (d) 2
32. 32.Which of these numbers is divisible by 11? (a) 2672 (b) 2673 (c) 2678 (d) 2692
33. 33.What is the units digit of 8^34? (a) 2 (b) 6 (c) 8 (d) 4
34. 34.Which of these numbers is divisible by 4? (a) 1505 (b) 1502 (c) 1511 (d) 1504
35. 35.What is the remainder when 570 is divided by 9? (a) 3 (b) 2 (c) 6 (d) 4
36. 36.What is the units digit of 8^11? (a) 2 (b) 6 (c) 4 (d) 8
37. 37.What is the smallest number you must add to 251 to make it divisible by 4? (a) 3 (b) 2 (c) 1
38. 38.Which of these numbers is divisible by 8? (a) 3963 (b) 3983 (c) 3970 (d) 3968
39. 39.Which of these numbers is divisible by 11? (a) 7691 (b) 7678 (c) 7687 (d) 7674
40. 40.What is the units digit of 8^15? (a) 8 (b) 4 (c) 2 (d) 6

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