Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 562 · 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.Which of these numbers is divisible by 3? (a) 2612 (b) 2609 (c) 2610 (d) 2615
2. 2.What is the smallest number you must add to 98 to make it divisible by 3? (a) 2 (b) 1
3. 3.What is the units digit of 2^54? (a) 8 (b) 6 (c) 4 (d) 2
4. 4.What is the smallest number you must add to 235 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
5. 5.Which of these numbers is divisible by 4? (a) 1634 (b) 1638 (c) 1636 (d) 1641
6. 6.What is the units digit of 3^29? (a) 7 (b) 1 (c) 3 (d) 9
7. 7.What is the units digit of 3^36? (a) 3 (b) 1 (c) 7 (d) 9
8. 8.What is the remainder when 254 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
9. 9.Which of these numbers is divisible by 8? (a) 5696 (b) 5690 (c) 5707 (d) 5702
10. 10.What is the remainder when 518 is divided by 9? (a) 4 (b) 6 (c) 7 (d) 5
11. 11.Which of these numbers is divisible by 11? (a) 1788 (b) 1794 (c) 1776 (d) 1782
12. 12.What is the smallest number you must add to 639 to make it divisible by 8? (a) 7 (b) 3 (c) 1 (d) 2
13. 13.What is the remainder when 574 is divided by 8? (a) 6 (b) 7 (c) 5 (d) 2
14. 14.What is the smallest number you must add to 219 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
15. 15.What is the remainder when 188 is divided by 3? (a) 0 (b) 2 (c) 1
16. 16.Which of these numbers is divisible by 6? (a) 2607 (b) 2594 (c) 2598 (d) 2601
17. 17.What is the smallest number you must add to 148 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
18. 18.What is the units digit of 3^35? (a) 3 (b) 9 (c) 1 (d) 7
19. 19.What is the units digit of 7^35? (a) 9 (b) 7 (c) 1 (d) 3
20. 20.What is the units digit of 2^47? (a) 2 (b) 4 (c) 6 (d) 8
21. 21.What is the remainder when 121 is divided by 3? (a) 2 (b) 0 (c) 1
22. 22.What is the units digit of 3^30? (a) 3 (b) 1 (c) 9 (d) 7
23. 23.What is the remainder when 395 is divided by 8? (a) 4 (b) 5 (c) 2 (d) 3
24. 24.What is the smallest number you must add to 279 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 6
25. 25.What is the smallest number you must add to 442 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
26. 26.What is the units digit of 2^37? (a) 8 (b) 2 (c) 4 (d) 6
27. 27.Which of these numbers is divisible by 6? (a) 4101 (b) 4092 (c) 4091 (d) 4093
28. 28.What is the remainder when 211 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
29. 29.Which of these numbers is divisible by 9? (a) 769 (b) 788 (c) 774 (d) 782
30. 30.What is the units digit of 7^12? (a) 9 (b) 3 (c) 1 (d) 7
31. 31.Which of these numbers is divisible by 3? (a) 1964 (b) 1967 (c) 1960 (d) 1962
32. 32.Which of these numbers is divisible by 3? (a) 772 (b) 776 (c) 774 (d) 779
33. 33.What is the smallest number you must add to 215 to make it divisible by 3? (a) 2 (b) 1
34. 34.What is the remainder when 107 is divided by 9? (a) 7 (b) 1 (c) 8 (d) 6
35. 35.Which of these numbers is divisible by 9? (a) 6599 (b) 6594 (c) 6597 (d) 6614
36. 36.Which of these numbers is divisible by 8? (a) 656 (b) 658 (c) 665 (d) 650
37. 37.What is the remainder when 695 is divided by 12? (a) 1 (b) 10 (c) 9 (d) 11
38. 38.Which of these numbers is divisible by 9? (a) 6056 (b) 6062 (c) 6057 (d) 6072
39. 39.What is the smallest number you must add to 317 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
40. 40.What is the remainder when 304 is divided by 10? (a) 3 (b) 6 (c) 5 (d) 4

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