Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 546 · 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 8? (a) 936 (b) 935 (c) 947 (d) 939
2. 2.What is the remainder when 326 is divided by 12? (a) 2 (b) 1 (c) 10 (d) 3
3. 3.What is the remainder when 644 is divided by 12? (a) 4 (b) 7 (c) 9 (d) 8
4. 4.Which of these numbers is divisible by 6? (a) 5400 (b) 5402 (c) 5397 (d) 5411
5. 5.Which of these numbers is divisible by 6? (a) 2710 (b) 2702 (c) 2713 (d) 2706
6. 6.What is the units digit of 4^46? (a) 6 (b) 5 (c) 7 (d) 4
7. 7.Which of these numbers is divisible by 11? (a) 8247 (b) 8228 (c) 8224 (d) 8229
8. 8.What is the remainder when 679 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 1
9. 9.What is the remainder when 355 is divided by 12? (a) 6 (b) 5 (c) 8 (d) 7
10. 10.Which of these numbers is divisible by 4? (a) 1941 (b) 1940 (c) 1945 (d) 1937
11. 11.What is the remainder when 614 is divided by 8? (a) 7 (b) 5 (c) 2 (d) 6
12. 12.What is the smallest number you must add to 137 to make it divisible by 3? (a) 1 (b) 2
13. 13.Which of these numbers is divisible by 8? (a) 3745 (b) 3766 (c) 3752 (d) 3758
14. 14.What is the smallest number you must add to 223 to make it divisible by 9? (a) 3 (b) 7 (c) 1 (d) 2
15. 15.Which of these numbers is divisible by 3? (a) 2475 (b) 2473 (c) 2477 (d) 2479
16. 16.Which of these numbers is divisible by 3? (a) 388 (b) 387 (c) 392 (d) 386
17. 17.What is the smallest number you must add to 346 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
18. 18.Which of these numbers is divisible by 11? (a) 6633 (b) 6624 (c) 6635 (d) 6652
19. 19.Which of these numbers is divisible by 3? (a) 1980 (b) 1979 (c) 1984 (d) 1982
20. 20.What is the units digit of 3^34? (a) 9 (b) 1 (c) 3 (d) 7
21. 21.What is the units digit of 4^59? (a) 5 (b) 3 (c) 6 (d) 4
22. 22.Which of these numbers is divisible by 9? (a) 5622 (b) 5613 (c) 5633 (d) 5616
23. 23.What is the smallest number you must add to 797 to make it divisible by 11? (a) 7 (b) 5 (c) 8 (d) 6
24. 24.What is the remainder when 205 is divided by 3? (a) 2 (b) 1 (c) 0
25. 25.Which of these numbers is divisible by 11? (a) 1666 (b) 1685 (c) 1672 (d) 1679
26. 26.What is the remainder when 411 is divided by 8? (a) 2 (b) 5 (c) 4 (d) 3
27. 27.Which of these numbers is divisible by 3? (a) 2414 (b) 2408 (c) 2409 (d) 2411
28. 28.What is the remainder when 657 is divided by 11? (a) 7 (b) 8 (c) 3 (d) 9
29. 29.What is the smallest number you must add to 188 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
30. 30.What is the remainder when 59 is divided by 3? (a) 1 (b) 2 (c) 0
31. 31.Which of these numbers is divisible by 8? (a) 2121 (b) 2116 (c) 2111 (d) 2112
32. 32.Which of these numbers is divisible by 8? (a) 5756 (b) 5760 (c) 5772 (d) 5765
33. 33.Which of these numbers is divisible by 3? (a) 1598 (b) 1600 (c) 1599 (d) 1603
34. 34.What is the smallest number you must add to 182 to make it divisible by 3? (a) 1 (b) 2
35. 35.What is the units digit of 9^44? (a) 2 (b) 9 (c) 1 (d) 0
36. 36.What is the units digit of 2^55? (a) 6 (b) 2 (c) 8 (d) 4
37. 37.What is the remainder when 202 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
38. 38.What is the units digit of 3^32? (a) 1 (b) 9 (c) 7 (d) 3
39. 39.Which of these numbers is divisible by 3? (a) 402 (b) 406 (c) 400 (d) 404
40. 40.What is the remainder when 784 is divided by 11? (a) 8 (b) 4 (c) 2 (d) 3

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