Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 455 · 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 smallest number you must add to 236 to make it divisible by 9? (a) 6 (b) 7 (c) 2 (d) 8
2. 2.What is the remainder when 970 is divided by 12? (a) 10 (b) 2 (c) 9 (d) 11
3. 3.What is the remainder when 67 is divided by 3? (a) 0 (b) 1 (c) 2
4. 4.What is the remainder when 201 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
5. 5.Which of these numbers is divisible by 4? (a) 1792 (b) 1794 (c) 1798 (d) 1790
6. 6.Which of these numbers is divisible by 3? (a) 1485 (b) 1490 (c) 1487 (d) 1484
7. 7.What is the remainder when 830 is divided by 12? (a) 3 (b) 10 (c) 2 (d) 1
8. 8.What is the remainder when 447 is divided by 10? (a) 6 (b) 8 (c) 7 (d) 3
9. 9.Which of these numbers is divisible by 9? (a) 4784 (b) 4777 (c) 4762 (d) 4770
10. 10.What is the smallest number you must add to 701 to make it divisible by 11? (a) 2 (b) 8 (c) 3 (d) 4
11. 11.What is the units digit of 7^34? (a) 1 (b) 9 (c) 7 (d) 3
12. 12.What is the units digit of 7^32? (a) 7 (b) 9 (c) 3 (d) 1
13. 13.Which of these numbers is divisible by 9? (a) 5049 (b) 5042 (c) 5063 (d) 5050
14. 14.What is the units digit of 4^19? (a) 3 (b) 5 (c) 4 (d) 6
15. 15.What is the units digit of 2^17? (a) 6 (b) 8 (c) 2 (d) 4
16. 16.Which of these numbers is divisible by 3? (a) 1391 (b) 1393 (c) 1389 (d) 1388
17. 17.What is the units digit of 3^56? (a) 7 (b) 3 (c) 1 (d) 9
18. 18.What is the smallest number you must add to 406 to make it divisible by 8? (a) 3 (b) 6 (c) 2 (d) 1
19. 19.What is the smallest number you must add to 193 to make it divisible by 11? (a) 4 (b) 5 (c) 7 (d) 6
20. 20.Which of these numbers is divisible by 11? (a) 2418 (b) 2421 (c) 2408 (d) 2409
21. 21.What is the units digit of 4^54? (a) 6 (b) 5 (c) 4 (d) 7
22. 22.What is the remainder when 284 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
23. 23.What is the units digit of 4^14? (a) 7 (b) 6 (c) 4 (d) 5
24. 24.Which of these numbers is divisible by 6? (a) 505 (b) 504 (c) 515 (d) 502
25. 25.What is the units digit of 2^10? (a) 6 (b) 2 (c) 8 (d) 4
26. 26.What is the units digit of 4^44? (a) 6 (b) 4 (c) 5 (d) 7
27. 27.What is the remainder when 242 is divided by 3? (a) 2 (b) 0 (c) 1
28. 28.What is the smallest number you must add to 142 to make it divisible by 4? (a) 1 (b) 3 (c) 2
29. 29.What is the remainder when 127 is divided by 3? (a) 2 (b) 0 (c) 1
30. 30.What is the smallest number you must add to 157 to make it divisible by 8? (a) 2 (b) 5 (c) 3 (d) 4
31. 31.What is the remainder when 42 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
32. 32.What is the smallest number you must add to 307 to make it divisible by 4? (a) 1 (b) 2 (c) 3
33. 33.Which of these numbers is divisible by 3? (a) 1918 (b) 1920 (c) 1924 (d) 1922
34. 34.Which of these numbers is divisible by 11? (a) 3734 (b) 3759 (c) 3746 (d) 3740
35. 35.What is the remainder when 204 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 8
36. 36.Which of these numbers is divisible by 3? (a) 707 (b) 703 (c) 710 (d) 705
37. 37.What is the units digit of 4^38? (a) 7 (b) 4 (c) 6 (d) 5
38. 38.Which of these numbers is divisible by 11? (a) 3745 (b) 3767 (c) 3757 (d) 3751
39. 39.Which of these numbers is divisible by 3? (a) 1463 (b) 1458 (c) 1456 (d) 1460
40. 40.Which of these numbers is divisible by 4? (a) 565 (b) 564 (c) 561 (d) 570

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