Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 463 · 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 611 is divided by 8? (a) 4 (b) 2 (c) 3 (d) 5
2. 2.What is the smallest number you must add to 490 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 2
3. 3.Which of these numbers is divisible by 9? (a) 4823 (b) 4813 (c) 4826 (d) 4815
4. 4.What is the remainder when 59 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
5. 5.What is the remainder when 172 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
6. 6.What is the units digit of 9^19? (a) 9 (b) 1 (c) 7 (d) 8
7. 7.What is the smallest number you must add to 114 to make it divisible by 4? (a) 1 (b) 2 (c) 3
8. 8.What is the smallest number you must add to 254 to make it divisible by 4? (a) 1 (b) 3 (c) 2
9. 9.What is the remainder when 898 is divided by 12? (a) 2 (b) 10 (c) 9 (d) 11
10. 10.What is the smallest number you must add to 735 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
11. 11.What is the smallest number you must add to 83 to make it divisible by 3? (a) 2 (b) 1
12. 12.What is the remainder when 659 is divided by 9? (a) 2 (b) 1 (c) 3 (d) 7
13. 13.What is the remainder when 682 is divided by 10? (a) 3 (b) 1 (c) 8 (d) 2
14. 14.What is the remainder when 653 is divided by 9? (a) 5 (b) 7 (c) 4 (d) 6
15. 15.Which of these numbers is divisible by 4? (a) 3543 (b) 3535 (c) 3536 (d) 3537
16. 16.What is the units digit of 4^17? (a) 3 (b) 4 (c) 5 (d) 6
17. 17.What is the remainder when 334 is divided by 11? (a) 3 (b) 4 (c) 5 (d) 7
18. 18.What is the units digit of 4^32? (a) 7 (b) 6 (c) 4 (d) 5
19. 19.What is the remainder when 668 is divided by 10? (a) 7 (b) 2 (c) 9 (d) 8
20. 20.What is the smallest number you must add to 88 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the remainder when 173 is divided by 9? (a) 3 (b) 1 (c) 2 (d) 7
22. 22.What is the units digit of 4^53? (a) 6 (b) 5 (c) 3 (d) 4
23. 23.Which of these numbers is divisible by 3? (a) 971 (b) 965 (c) 968 (d) 966
24. 24.Which of these numbers is divisible by 9? (a) 4336 (b) 4319 (c) 4322 (d) 4320
25. 25.What is the units digit of 9^21? (a) 7 (b) 1 (c) 9 (d) 8
26. 26.Which of these numbers is divisible by 9? (a) 1230 (b) 1220 (c) 1236 (d) 1224
27. 27.What is the units digit of 7^52? (a) 3 (b) 1 (c) 9 (d) 7
28. 28.Which of these numbers is divisible by 6? (a) 633 (b) 629 (c) 638 (d) 630
29. 29.Which of these numbers is divisible by 11? (a) 3741 (b) 3737 (c) 3740 (d) 3761
30. 30.Which of these numbers is divisible by 3? (a) 1892 (b) 1894 (c) 1889 (d) 1890
31. 31.What is the remainder when 186 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
32. 32.Which of these numbers is divisible by 3? (a) 2585 (b) 2590 (c) 2588 (d) 2586
33. 33.What is the remainder when 95 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 6
34. 34.What is the remainder when 72 is divided by 7? (a) 2 (b) 1 (c) 3 (d) 5
35. 35.What is the remainder when 749 is divided by 12? (a) 5 (b) 4 (c) 6 (d) 7
36. 36.Which of these numbers is divisible by 9? (a) 7440 (b) 7448 (c) 7434 (d) 7427
37. 37.What is the units digit of 2^56? (a) 4 (b) 6 (c) 8 (d) 2
38. 38.Which of these numbers is divisible by 11? (a) 3193 (b) 3180 (c) 3173 (d) 3179
39. 39.What is the smallest number you must add to 139 to make it divisible by 4? (a) 3 (b) 2 (c) 1
40. 40.What is the smallest number you must add to 74 to make it divisible by 3? (a) 2 (b) 1

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