Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 28 · 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 units digit of 7^25? (a) 9 (b) 7 (c) 1 (d) 3
2. 2.What is the units digit of 2^55? (a) 2 (b) 4 (c) 8 (d) 6
3. 3.What is the remainder when 750 is divided by 12? (a) 5 (b) 7 (c) 8 (d) 6
4. 4.Which of these numbers is divisible by 3? (a) 1454 (b) 1455 (c) 1457 (d) 1459
5. 5.Which of these numbers is divisible by 3? (a) 1775 (b) 1780 (c) 1777 (d) 1776
6. 6.What is the units digit of 3^31? (a) 7 (b) 3 (c) 9 (d) 1
7. 7.Which of these numbers is divisible by 6? (a) 4227 (b) 4222 (c) 4231 (d) 4224
8. 8.What is the remainder when 659 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2
9. 9.Which of these numbers is divisible by 4? (a) 3310 (b) 3303 (c) 3304 (d) 3305
10. 10.What is the remainder when 466 is divided by 11? (a) 3 (b) 4 (c) 7 (d) 5
11. 11.What is the units digit of 3^17? (a) 3 (b) 9 (c) 1 (d) 7
12. 12.What is the units digit of 7^46? (a) 3 (b) 1 (c) 9 (d) 7
13. 13.What is the units digit of 8^17? (a) 8 (b) 2 (c) 6 (d) 4
14. 14.Which of these numbers is divisible by 4? (a) 1033 (b) 1028 (c) 1031 (d) 1025
15. 15.What is the units digit of 3^46? (a) 7 (b) 3 (c) 1 (d) 9
16. 16.What is the smallest number you must add to 698 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 2
17. 17.What is the units digit of 8^21? (a) 4 (b) 2 (c) 8 (d) 6
18. 18.What is the units digit of 3^19? (a) 7 (b) 3 (c) 9 (d) 1
19. 19.Which of these numbers is divisible by 4? (a) 232 (b) 238 (c) 230 (d) 234
20. 20.Which of these numbers is divisible by 4? (a) 3445 (b) 3442 (c) 3444 (d) 3449
21. 21.Which of these numbers is divisible by 3? (a) 2485 (b) 2483 (c) 2481 (d) 2480
22. 22.Which of these numbers is divisible by 3? (a) 941 (b) 938 (c) 943 (d) 939
23. 23.What is the remainder when 179 is divided by 11? (a) 8 (b) 4 (c) 2 (d) 3
24. 24.What is the units digit of 2^54? (a) 4 (b) 2 (c) 8 (d) 6
25. 25.What is the smallest number you must add to 475 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
26. 26.Which of these numbers is divisible by 8? (a) 5265 (b) 5256 (c) 5255 (d) 5263
27. 27.What is the remainder when 165 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
28. 28.Which of these numbers is divisible by 3? (a) 1025 (b) 1026 (c) 1031 (d) 1027
29. 29.What is the smallest number you must add to 404 to make it divisible by 7? (a) 3 (b) 1 (c) 5 (d) 2
30. 30.Which of these numbers is divisible by 11? (a) 8027 (b) 8048 (c) 8030 (d) 8034
31. 31.What is the smallest number you must add to 500 to make it divisible by 11? (a) 7 (b) 5 (c) 8 (d) 6
32. 32.Which of these numbers is divisible by 8? (a) 2176 (b) 2185 (c) 2179 (d) 2173
33. 33.What is the units digit of 3^14? (a) 7 (b) 1 (c) 3 (d) 9
34. 34.What is the smallest number you must add to 376 to make it divisible by 11? (a) 8 (b) 10 (c) 9 (d) 2
35. 35.Which of these numbers is divisible by 8? (a) 1966 (b) 1974 (c) 1968 (d) 1982
36. 36.What is the smallest number you must add to 239 to make it divisible by 8? (a) 2 (b) 3 (c) 7 (d) 1
37. 37.What is the smallest number you must add to 290 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 2
38. 38.What is the smallest number you must add to 408 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 8
39. 39.What is the smallest number you must add to 469 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 3
40. 40.What is the smallest number you must add to 217 to make it divisible by 3? (a) 1 (b) 2

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