Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 167 · 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 256 is divided by 11? (a) 8 (b) 3 (c) 2 (d) 4
2. 2.What is the units digit of 7^28? (a) 9 (b) 7 (c) 3 (d) 1
3. 3.What is the remainder when 341 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
4. 4.What is the smallest number you must add to 553 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 4
5. 5.What is the units digit of 8^27? (a) 6 (b) 8 (c) 2 (d) 4
6. 6.What is the remainder when 404 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
7. 7.What is the units digit of 9^14? (a) 0 (b) 1 (c) 2 (d) 9
8. 8.What is the remainder when 116 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
9. 9.Which of these numbers is divisible by 8? (a) 4532 (b) 4522 (c) 4537 (d) 4528
10. 10.What is the remainder when 681 is divided by 9? (a) 7 (b) 3 (c) 6 (d) 5
11. 11.What is the remainder when 129 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 4
12. 12.What is the remainder when 124 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 2
13. 13.Which of these numbers is divisible by 6? (a) 4812 (b) 4816 (c) 4819 (d) 4810
14. 14.Which of these numbers is divisible by 9? (a) 7850 (b) 7844 (c) 7839 (d) 7831
15. 15.What is the remainder when 76 is divided by 7? (a) 1 (b) 6 (c) 5 (d) 4
16. 16.What is the smallest number you must add to 155 to make it divisible by 10? (a) 6 (b) 7 (c) 4 (d) 5
17. 17.Which of these numbers is divisible by 8? (a) 1976 (b) 1983 (c) 1988 (d) 1975
18. 18.What is the remainder when 245 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
19. 19.What is the units digit of 4^18? (a) 6 (b) 7 (c) 5 (d) 4
20. 20.What is the units digit of 4^45? (a) 5 (b) 4 (c) 3 (d) 6
21. 21.What is the smallest number you must add to 749 to make it divisible by 10? (a) 3 (b) 9 (c) 2 (d) 1
22. 22.Which of these numbers is divisible by 11? (a) 1606 (b) 1614 (c) 1602 (d) 1620
23. 23.What is the smallest number you must add to 142 to make it divisible by 7? (a) 6 (b) 5 (c) 2 (d) 4
24. 24.What is the units digit of 4^23? (a) 4 (b) 5 (c) 6 (d) 3
25. 25.Which of these numbers is divisible by 6? (a) 1213 (b) 1204 (c) 1206 (d) 1207
26. 26.What is the units digit of 3^21? (a) 7 (b) 3 (c) 1 (d) 9
27. 27.Which of these numbers is divisible by 4? (a) 3376 (b) 3374 (c) 3382 (d) 3379
28. 28.What is the units digit of 2^51? (a) 2 (b) 8 (c) 6 (d) 4
29. 29.What is the smallest number you must add to 403 to make it divisible by 10? (a) 6 (b) 3 (c) 8 (d) 7
30. 30.What is the units digit of 4^43? (a) 4 (b) 6 (c) 5 (d) 3
31. 31.What is the remainder when 195 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
32. 32.What is the remainder when 400 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 5
33. 33.What is the remainder when 609 is divided by 9? (a) 5 (b) 7 (c) 3 (d) 6
34. 34.What is the smallest number you must add to 416 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
35. 35.Which of these numbers is divisible by 4? (a) 1493 (b) 1490 (c) 1498 (d) 1492
36. 36.What is the remainder when 855 is divided by 12? (a) 4 (b) 3 (c) 2 (d) 9
37. 37.Which of these numbers is divisible by 3? (a) 461 (b) 458 (c) 454 (d) 456
38. 38.Which of these numbers is divisible by 3? (a) 2453 (b) 2456 (c) 2451 (d) 2449
39. 39.What is the smallest number you must add to 668 to make it divisible by 9? (a) 8 (b) 2 (c) 6 (d) 7
40. 40.What is the units digit of 2^12? (a) 4 (b) 8 (c) 2 (d) 6

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