Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 424 · 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 508 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 3
2. 2.What is the remainder when 307 is divided by 8? (a) 3 (b) 4 (c) 2 (d) 5
3. 3.What is the units digit of 3^40? (a) 7 (b) 9 (c) 1 (d) 3
4. 4.Which of these numbers is divisible by 8? (a) 6191 (b) 6182 (c) 6172 (d) 6176
5. 5.What is the remainder when 187 is divided by 3? (a) 1 (b) 0 (c) 2
6. 6.What is the remainder when 351 is divided by 12? (a) 2 (b) 4 (c) 3 (d) 9
7. 7.Which of these numbers is divisible by 8? (a) 5676 (b) 5681 (c) 5692 (d) 5680
8. 8.What is the smallest number you must add to 403 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 6
9. 9.What is the units digit of 9^60? (a) 2 (b) 9 (c) 0 (d) 1
10. 10.What is the units digit of 8^24? (a) 4 (b) 2 (c) 8 (d) 6
11. 11.Which of these numbers is divisible by 11? (a) 8525 (b) 8523 (c) 8539 (d) 8526
12. 12.What is the smallest number you must add to 91 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
13. 13.Which of these numbers is divisible by 3? (a) 809 (b) 806 (c) 812 (d) 807
14. 14.What is the remainder when 159 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
15. 15.What is the units digit of 4^17? (a) 3 (b) 6 (c) 5 (d) 4
16. 16.What is the smallest number you must add to 362 to make it divisible by 11? (a) 2 (b) 1 (c) 10 (d) 3
17. 17.What is the units digit of 7^24? (a) 3 (b) 9 (c) 1 (d) 7
18. 18.What is the smallest number you must add to 698 to make it divisible by 9? (a) 5 (b) 3 (c) 4 (d) 6
19. 19.Which of these numbers is divisible by 11? (a) 4665 (b) 4693 (c) 4675 (d) 4681
20. 20.What is the remainder when 262 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
21. 21.What is the smallest number you must add to 58 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
22. 22.What is the remainder when 113 is divided by 3? (a) 0 (b) 1 (c) 2
23. 23.Which of these numbers is divisible by 3? (a) 2598 (b) 2599 (c) 2603 (d) 2597
24. 24.What is the smallest number you must add to 189 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
25. 25.Which of these numbers is divisible by 11? (a) 2934 (b) 2956 (c) 2945 (d) 2937
26. 26.What is the smallest number you must add to 215 to make it divisible by 3? (a) 1 (b) 2
27. 27.Which of these numbers is divisible by 8? (a) 5739 (b) 5736 (c) 5734 (d) 5748
28. 28.What is the smallest number you must add to 220 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 4
29. 29.Which of these numbers is divisible by 8? (a) 4641 (b) 4640 (c) 4638 (d) 4652
30. 30.What is the smallest number you must add to 413 to make it divisible by 6? (a) 5 (b) 1 (c) 2 (d) 3
31. 31.Which of these numbers is divisible by 9? (a) 6758 (b) 6759 (c) 6773 (d) 6767
32. 32.What is the smallest number you must add to 339 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
33. 33.What is the units digit of 4^22? (a) 7 (b) 4 (c) 6 (d) 5
34. 34.What is the units digit of 3^30? (a) 7 (b) 3 (c) 1 (d) 9
35. 35.What is the remainder when 249 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 3
36. 36.What is the remainder when 203 is divided by 10? (a) 3 (b) 4 (c) 7 (d) 2
37. 37.What is the units digit of 8^52? (a) 8 (b) 4 (c) 6 (d) 2
38. 38.What is the remainder when 660 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
39. 39.Which of these numbers is divisible by 3? (a) 884 (b) 881 (c) 886 (d) 882
40. 40.What is the remainder when 208 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5

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