Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 835 · 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 162 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
2. 2.What is the smallest number you must add to 617 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
3. 3.What is the smallest number you must add to 749 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 8
4. 4.What is the smallest number you must add to 302 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
5. 5.What is the units digit of 2^22? (a) 8 (b) 2 (c) 4 (d) 6
6. 6.What is the units digit of 9^33? (a) 1 (b) 8 (c) 7 (d) 9
7. 7.What is the remainder when 422 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
8. 8.Which of these numbers is divisible by 9? (a) 1963 (b) 1971 (c) 1973 (d) 1986
9. 9.What is the units digit of 2^14? (a) 2 (b) 8 (c) 6 (d) 4
10. 10.What is the remainder when 881 is divided by 11? (a) 2 (b) 1 (c) 0 (d) 10
11. 11.What is the smallest number you must add to 66 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
12. 12.Which of these numbers is divisible by 11? (a) 6116 (b) 6132 (c) 6115 (d) 6119
13. 13.Which of these numbers is divisible by 8? (a) 1208 (b) 1215 (c) 1222 (d) 1203
14. 14.Which of these numbers is divisible by 6? (a) 4191 (b) 4198 (c) 4184 (d) 4188
15. 15.What is the units digit of 8^37? (a) 6 (b) 2 (c) 4 (d) 8
16. 16.Which of these numbers is divisible by 11? (a) 2649 (b) 2629 (c) 2636 (d) 2626
17. 17.Which of these numbers is divisible by 8? (a) 2924 (b) 2934 (c) 2920 (d) 2919
18. 18.What is the smallest number you must add to 83 to make it divisible by 3? (a) 2 (b) 1
19. 19.What is the remainder when 214 is divided by 10? (a) 5 (b) 3 (c) 6 (d) 4
20. 20.What is the remainder when 113 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
21. 21.What is the remainder when 520 is divided by 9? (a) 8 (b) 6 (c) 2 (d) 7
22. 22.Which of these numbers is divisible by 4? (a) 1751 (b) 1754 (c) 1745 (d) 1748
23. 23.What is the units digit of 4^43? (a) 6 (b) 5 (c) 4 (d) 3
24. 24.What is the remainder when 114 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
25. 25.Which of these numbers is divisible by 4? (a) 3489 (b) 3488 (c) 3487 (d) 3493
26. 26.What is the units digit of 7^35? (a) 9 (b) 7 (c) 1 (d) 3
27. 27.What is the units digit of 9^37? (a) 9 (b) 1 (c) 7 (d) 8
28. 28.What is the smallest number you must add to 267 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
29. 29.What is the units digit of 4^60? (a) 7 (b) 4 (c) 6 (d) 5
30. 30.Which of these numbers is divisible by 9? (a) 3790 (b) 3780 (c) 3786 (d) 3772
31. 31.What is the remainder when 431 is divided by 7? (a) 6 (b) 4 (c) 3 (d) 5
32. 32.What is the units digit of 3^37? (a) 3 (b) 7 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 9? (a) 855 (b) 851 (c) 865 (d) 859
34. 34.Which of these numbers is divisible by 6? (a) 1147 (b) 1144 (c) 1146 (d) 1153
35. 35.What is the smallest number you must add to 479 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 1
36. 36.What is the units digit of 8^42? (a) 8 (b) 4 (c) 6 (d) 2
37. 37.Which of these numbers is divisible by 8? (a) 5595 (b) 5600 (c) 5614 (d) 5605
38. 38.What is the remainder when 161 is divided by 8? (a) 0 (b) 2 (c) 7 (d) 1
39. 39.What is the units digit of 2^15? (a) 2 (b) 8 (c) 6 (d) 4
40. 40.What is the remainder when 490 is divided by 9? (a) 3 (b) 5 (c) 4 (d) 6

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