Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 898 · 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 219 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
2. 2.What is the units digit of 7^45? (a) 1 (b) 7 (c) 3 (d) 9
3. 3.What is the units digit of 7^55? (a) 7 (b) 3 (c) 1 (d) 9
4. 4.Which of these numbers is divisible by 11? (a) 9581 (b) 9589 (c) 9579 (d) 9597
5. 5.What is the smallest number you must add to 549 to make it divisible by 7? (a) 6 (b) 3 (c) 5 (d) 4
6. 6.What is the units digit of 2^12? (a) 4 (b) 6 (c) 2 (d) 8
7. 7.What is the units digit of 7^15? (a) 3 (b) 1 (c) 9 (d) 7
8. 8.Which of these numbers is divisible by 11? (a) 3482 (b) 3494 (c) 3505 (d) 3487
9. 9.What is the smallest number you must add to 923 to make it divisible by 12? (a) 2 (b) 1 (c) 3 (d) 11
10. 10.Which of these numbers is divisible by 8? (a) 5724 (b) 5713 (c) 5720 (d) 5729
11. 11.What is the remainder when 217 is divided by 8? (a) 7 (b) 0 (c) 1 (d) 2
12. 12.What is the units digit of 7^27? (a) 9 (b) 1 (c) 3 (d) 7
13. 13.What is the units digit of 3^53? (a) 7 (b) 3 (c) 9 (d) 1
14. 14.Which of these numbers is divisible by 4? (a) 2831 (b) 2821 (c) 2824 (d) 2825
15. 15.What is the units digit of 4^50? (a) 4 (b) 5 (c) 7 (d) 6
16. 16.What is the remainder when 143 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
17. 17.Which of these numbers is divisible by 9? (a) 5908 (b) 5904 (c) 5896 (d) 5914
18. 18.What is the remainder when 248 is divided by 11? (a) 6 (b) 8 (c) 7 (d) 5
19. 19.What is the smallest number you must add to 43 to make it divisible by 3? (a) 1 (b) 2
20. 20.What is the units digit of 8^14? (a) 6 (b) 8 (c) 4 (d) 2
21. 21.What is the smallest number you must add to 186 to make it divisible by 4? (a) 1 (b) 3 (c) 2
22. 22.What is the remainder when 358 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
23. 23.What is the remainder when 457 is divided by 12? (a) 11 (b) 2 (c) 1 (d) 0
24. 24.What is the remainder when 315 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
25. 25.Which of these numbers is divisible by 3? (a) 1761 (b) 1759 (c) 1762 (d) 1765
26. 26.What is the units digit of 3^50? (a) 7 (b) 1 (c) 3 (d) 9
27. 27.Which of these numbers is divisible by 9? (a) 3525 (b) 3503 (c) 3510 (d) 3511
28. 28.Which of these numbers is divisible by 4? (a) 3101 (b) 3095 (c) 3098 (d) 3096
29. 29.What is the smallest number you must add to 114 to make it divisible by 11? (a) 8 (b) 6 (c) 7 (d) 4
30. 30.What is the units digit of 8^40? (a) 6 (b) 2 (c) 4 (d) 8
31. 31.Which of these numbers is divisible by 11? (a) 5336 (b) 5355 (c) 5330 (d) 5335
32. 32.What is the units digit of 3^17? (a) 1 (b) 3 (c) 7 (d) 9
33. 33.What is the units digit of 9^37? (a) 8 (b) 9 (c) 7 (d) 1
34. 34.What is the smallest number you must add to 78 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
35. 35.What is the units digit of 4^42? (a) 6 (b) 7 (c) 4 (d) 5
36. 36.What is the units digit of 2^51? (a) 2 (b) 8 (c) 4 (d) 6
37. 37.What is the units digit of 2^28? (a) 8 (b) 2 (c) 4 (d) 6
38. 38.What is the smallest number you must add to 324 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
39. 39.What is the smallest number you must add to 151 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 157 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3

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