Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 852 · 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 3^14? (a) 3 (b) 9 (c) 7 (d) 1
2. 2.What is the units digit of 4^11? (a) 5 (b) 3 (c) 6 (d) 4
3. 3.Which of these numbers is divisible by 9? (a) 761 (b) 778 (c) 765 (d) 773
4. 4.What is the smallest number you must add to 103 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 7
5. 5.What is the units digit of 7^60? (a) 1 (b) 3 (c) 9 (d) 7
6. 6.Which of these numbers is divisible by 8? (a) 4373 (b) 4391 (c) 4383 (d) 4376
7. 7.Which of these numbers is divisible by 11? (a) 8775 (b) 8797 (c) 8783 (d) 8778
8. 8.What is the smallest number you must add to 475 to make it divisible by 8? (a) 3 (b) 5 (c) 6 (d) 4
9. 9.What is the remainder when 443 is divided by 12? (a) 9 (b) 11 (c) 1 (d) 10
10. 10.Which of these numbers is divisible by 4? (a) 311 (b) 307 (c) 301 (d) 304
11. 11.Which of these numbers is divisible by 6? (a) 4637 (b) 4638 (c) 4648 (d) 4640
12. 12.What is the smallest number you must add to 175 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the smallest number you must add to 182 to make it divisible by 4? (a) 2 (b) 3 (c) 1
14. 14.What is the remainder when 74 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
15. 15.Which of these numbers is divisible by 4? (a) 2717 (b) 2709 (c) 2712 (d) 2714
16. 16.What is the units digit of 2^15? (a) 6 (b) 2 (c) 8 (d) 4
17. 17.Which of these numbers is divisible by 4? (a) 487 (b) 478 (c) 480 (d) 482
18. 18.What is the remainder when 204 is divided by 9? (a) 3 (b) 7 (c) 5 (d) 6
19. 19.What is the smallest number you must add to 310 to make it divisible by 4? (a) 3 (b) 1 (c) 2
20. 20.What is the units digit of 3^45? (a) 9 (b) 1 (c) 3 (d) 7
21. 21.What is the units digit of 2^27? (a) 2 (b) 4 (c) 6 (d) 8
22. 22.What is the remainder when 141 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 7
23. 23.What is the units digit of 2^18? (a) 2 (b) 6 (c) 4 (d) 8
24. 24.What is the smallest number you must add to 129 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
25. 25.What is the remainder when 507 is divided by 9? (a) 4 (b) 6 (c) 2 (d) 3
26. 26.What is the remainder when 139 is divided by 3? (a) 1 (b) 2 (c) 0
27. 27.Which of these numbers is divisible by 3? (a) 1121 (b) 1123 (c) 1117 (d) 1119
28. 28.Which of these numbers is divisible by 8? (a) 543 (b) 532 (c) 545 (d) 536
29. 29.What is the smallest number you must add to 406 to make it divisible by 12? (a) 3 (b) 10 (c) 2 (d) 1
30. 30.What is the smallest number you must add to 115 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
31. 31.What is the smallest number you must add to 63 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
32. 32.Which of these numbers is divisible by 11? (a) 4928 (b) 4940 (c) 4933 (d) 4924
33. 33.Which of these numbers is divisible by 8? (a) 1202 (b) 1221 (c) 1212 (d) 1208
34. 34.What is the remainder when 434 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
35. 35.What is the remainder when 291 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
36. 36.What is the units digit of 8^55? (a) 2 (b) 6 (c) 4 (d) 8
37. 37.What is the units digit of 4^54? (a) 4 (b) 7 (c) 6 (d) 5
38. 38.What is the remainder when 187 is divided by 8? (a) 4 (b) 2 (c) 5 (d) 3
39. 39.What is the smallest number you must add to 523 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
40. 40.Which of these numbers is divisible by 11? (a) 4917 (b) 4910 (c) 4934 (d) 4927

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