Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 618 · 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 232 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6
2. 2.What is the remainder when 619 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 1947 (b) 1949 (c) 1953 (d) 1948
4. 4.What is the remainder when 506 is divided by 9? (a) 2 (b) 7 (c) 1 (d) 3
5. 5.What is the units digit of 2^40? (a) 4 (b) 2 (c) 6 (d) 8
6. 6.What is the smallest number you must add to 172 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
7. 7.What is the units digit of 8^36? (a) 6 (b) 8 (c) 2 (d) 4
8. 8.What is the remainder when 713 is divided by 11? (a) 8 (b) 9 (c) 2 (d) 10
9. 9.Which of these numbers is divisible by 8? (a) 415 (b) 417 (c) 405 (d) 408
10. 10.Which of these numbers is divisible by 11? (a) 8797 (b) 8809 (c) 8800 (d) 8814
11. 11.Which of these numbers is divisible by 8? (a) 4428 (b) 4424 (c) 4417 (d) 4435
12. 12.Which of these numbers is divisible by 6? (a) 2504 (b) 2501 (c) 2502 (d) 2511
13. 13.What is the remainder when 418 is divided by 12? (a) 10 (b) 9 (c) 11 (d) 2
14. 14.What is the remainder when 176 is divided by 7? (a) 1 (b) 6 (c) 0 (d) 2
15. 15.What is the smallest number you must add to 54 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
16. 16.Which of these numbers is divisible by 3? (a) 1506 (b) 1505 (c) 1510 (d) 1508
17. 17.What is the smallest number you must add to 85 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
18. 18.What is the smallest number you must add to 247 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
19. 19.Which of these numbers is divisible by 6? (a) 2776 (b) 2779 (c) 2786 (d) 2778
20. 20.Which of these numbers is divisible by 6? (a) 5225 (b) 5218 (c) 5220 (d) 5228
21. 21.What is the smallest number you must add to 615 to make it divisible by 12? (a) 8 (b) 3 (c) 9 (d) 10
22. 22.What is the smallest number you must add to 68 to make it divisible by 3? (a) 1 (b) 2
23. 23.What is the smallest number you must add to 196 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the units digit of 4^13? (a) 6 (b) 3 (c) 5 (d) 4
25. 25.Which of these numbers is divisible by 8? (a) 568 (b) 583 (c) 570 (d) 561
26. 26.What is the remainder when 406 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
27. 27.What is the remainder when 738 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
28. 28.What is the units digit of 2^18? (a) 8 (b) 6 (c) 2 (d) 4
29. 29.What is the smallest number you must add to 49 to make it divisible by 4? (a) 3 (b) 1 (c) 2
30. 30.What is the remainder when 172 is divided by 3? (a) 2 (b) 1 (c) 0
31. 31.Which of these numbers is divisible by 8? (a) 1843 (b) 1834 (c) 1832 (d) 1831
32. 32.What is the smallest number you must add to 489 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
33. 33.What is the remainder when 456 is divided by 7? (a) 6 (b) 1 (c) 2 (d) 0
34. 34.What is the units digit of 3^20? (a) 7 (b) 9 (c) 1 (d) 3
35. 35.Which of these numbers is divisible by 11? (a) 8177 (b) 8152 (c) 8172 (d) 8162
36. 36.What is the smallest number you must add to 59 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
37. 37.What is the units digit of 2^20? (a) 8 (b) 4 (c) 6 (d) 2
38. 38.What is the smallest number you must add to 878 to make it divisible by 12? (a) 2 (b) 11 (c) 9 (d) 10
39. 39.What is the smallest number you must add to 58 to make it divisible by 3? (a) 1 (b) 2
40. 40.Which of these numbers is divisible by 11? (a) 807 (b) 796 (c) 819 (d) 803

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