Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 715 · 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 9? (a) 8108 (b) 8099 (c) 8087 (d) 8091
2. 2.What is the units digit of 8^46? (a) 2 (b) 8 (c) 6 (d) 4
3. 3.What is the remainder when 509 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
4. 4.Which of these numbers is divisible by 11? (a) 2555 (b) 2565 (c) 2552 (d) 2548
5. 5.What is the units digit of 7^53? (a) 7 (b) 3 (c) 9 (d) 1
6. 6.What is the units digit of 9^21? (a) 1 (b) 9 (c) 8 (d) 7
7. 7.What is the units digit of 7^20? (a) 9 (b) 7 (c) 3 (d) 1
8. 8.What is the smallest number you must add to 343 to make it divisible by 12? (a) 7 (b) 4 (c) 6 (d) 5
9. 9.What is the remainder when 339 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
10. 10.What is the smallest number you must add to 145 to make it divisible by 7? (a) 5 (b) 2 (c) 1 (d) 3
11. 11.Which of these numbers is divisible by 11? (a) 4994 (b) 4989 (c) 5002 (d) 5013
12. 12.What is the smallest number you must add to 805 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 1
13. 13.What is the units digit of 7^37? (a) 3 (b) 9 (c) 1 (d) 7
14. 14.What is the units digit of 4^51? (a) 6 (b) 5 (c) 4 (d) 3
15. 15.Which of these numbers is divisible by 9? (a) 5121 (b) 5120 (c) 5132 (d) 5125
16. 16.Which of these numbers is divisible by 4? (a) 3270 (b) 3272 (c) 3273 (d) 3277
17. 17.What is the units digit of 7^12? (a) 9 (b) 1 (c) 3 (d) 7
18. 18.Which of these numbers is divisible by 11? (a) 4643 (b) 4661 (c) 4653 (d) 4665
19. 19.What is the remainder when 424 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5
20. 20.Which of these numbers is divisible by 8? (a) 4665 (b) 4672 (c) 4677 (d) 4686
21. 21.Which of these numbers is divisible by 8? (a) 2153 (b) 2163 (c) 2160 (d) 2173
22. 22.What is the remainder when 149 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
23. 23.Which of these numbers is divisible by 3? (a) 2047 (b) 2043 (c) 2041 (d) 2044
24. 24.What is the units digit of 2^33? (a) 2 (b) 6 (c) 4 (d) 8
25. 25.What is the smallest number you must add to 557 to make it divisible by 8? (a) 4 (b) 2 (c) 5 (d) 3
26. 26.What is the remainder when 732 is divided by 10? (a) 2 (b) 3 (c) 1 (d) 8
27. 27.What is the remainder when 114 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
28. 28.What is the smallest number you must add to 203 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the smallest number you must add to 49 to make it divisible by 3? (a) 2 (b) 1
30. 30.What is the smallest number you must add to 386 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5
31. 31.Which of these numbers is divisible by 3? (a) 137 (b) 133 (c) 139 (d) 135
32. 32.What is the smallest number you must add to 461 to make it divisible by 7? (a) 6 (b) 2 (c) 1 (d) 3
33. 33.Which of these numbers is divisible by 3? (a) 664 (b) 659 (c) 660 (d) 661
34. 34.What is the units digit of 8^36? (a) 2 (b) 8 (c) 4 (d) 6
35. 35.What is the smallest number you must add to 473 to make it divisible by 10? (a) 8 (b) 6 (c) 3 (d) 7
36. 36.What is the units digit of 7^29? (a) 1 (b) 7 (c) 9 (d) 3
37. 37.What is the smallest number you must add to 495 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3
38. 38.What is the smallest number you must add to 284 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 4
39. 39.What is the smallest number you must add to 757 to make it divisible by 11? (a) 2 (b) 3 (c) 9 (d) 1
40. 40.Which of these numbers is divisible by 8? (a) 1571 (b) 1565 (c) 1568 (d) 1582

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