Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 6 · 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 units digit of 9^43? (a) 8 (b) 9 (c) 7 (d) 1
2. 2.Which of these numbers is divisible by 11? (a) 4992 (b) 4997 (c) 4982 (d) 4983
3. 3.What is the remainder when 110 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 1
4. 4.What is the smallest number you must add to 756 to make it divisible by 11? (a) 3 (b) 2 (c) 8 (d) 4
5. 5.What is the remainder when 126 is divided by 8? (a) 6 (b) 2 (c) 5 (d) 7
6. 6.What is the remainder when 289 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
7. 7.What is the units digit of 3^34? (a) 7 (b) 3 (c) 1 (d) 9
8. 8.Which of these numbers is divisible by 4? (a) 1642 (b) 1644 (c) 1647 (d) 1650
9. 9.What is the remainder when 359 is divided by 9? (a) 7 (b) 1 (c) 6 (d) 8
10. 10.What is the smallest number you must add to 376 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
11. 11.What is the smallest number you must add to 371 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
12. 12.What is the smallest number you must add to 634 to make it divisible by 12? (a) 3 (b) 2 (c) 10 (d) 1
13. 13.Which of these numbers is divisible by 6? (a) 4127 (b) 4122 (c) 4129 (d) 4117
14. 14.What is the units digit of 7^52? (a) 1 (b) 9 (c) 7 (d) 3
15. 15.Which of these numbers is divisible by 8? (a) 5963 (b) 5977 (c) 5970 (d) 5968
16. 16.What is the smallest number you must add to 167 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the units digit of 9^25? (a) 7 (b) 8 (c) 1 (d) 9
18. 18.What is the units digit of 3^24? (a) 1 (b) 3 (c) 7 (d) 9
19. 19.Which of these numbers is divisible by 11? (a) 3058 (b) 3061 (c) 3057 (d) 3072
20. 20.What is the units digit of 3^50? (a) 7 (b) 3 (c) 1 (d) 9
21. 21.Which of these numbers is divisible by 11? (a) 2866 (b) 2857 (c) 2845 (d) 2849
22. 22.What is the remainder when 242 is divided by 9? (a) 8 (b) 6 (c) 7 (d) 1
23. 23.What is the units digit of 8^31? (a) 6 (b) 4 (c) 2 (d) 8
24. 24.What is the remainder when 399 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
25. 25.Which of these numbers is divisible by 11? (a) 6445 (b) 6423 (c) 6434 (d) 6424
26. 26.Which of these numbers is divisible by 11? (a) 1135 (b) 1164 (c) 1144 (d) 1150
27. 27.Which of these numbers is divisible by 8? (a) 3106 (b) 3116 (c) 3104 (d) 3099
28. 28.What is the remainder when 573 is divided by 12? (a) 10 (b) 8 (c) 3 (d) 9
29. 29.What is the units digit of 7^19? (a) 1 (b) 7 (c) 3 (d) 9
30. 30.What is the smallest number you must add to 414 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
31. 31.What is the units digit of 2^22? (a) 6 (b) 2 (c) 4 (d) 8
32. 32.What is the remainder when 133 is divided by 8? (a) 6 (b) 5 (c) 4 (d) 3
33. 33.Which of these numbers is divisible by 4? (a) 1113 (b) 1109 (c) 1112 (d) 1117
34. 34.Which of these numbers is divisible by 9? (a) 4896 (b) 4894 (c) 4898 (d) 4912
35. 35.What is the units digit of 8^50? (a) 2 (b) 6 (c) 8 (d) 4
36. 36.What is the remainder when 101 is divided by 3? (a) 2 (b) 1 (c) 0
37. 37.What is the remainder when 878 is divided by 11? (a) 2 (b) 9 (c) 10 (d) 8
38. 38.What is the units digit of 8^47? (a) 6 (b) 4 (c) 2 (d) 8
39. 39.What is the remainder when 784 is divided by 10? (a) 5 (b) 3 (c) 4 (d) 6
40. 40.Which of these numbers is divisible by 11? (a) 4557 (b) 4545 (c) 4534 (d) 4543

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