Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 252 · 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 243 is divided by 12? (a) 9 (b) 3 (c) 2 (d) 4
2. 2.What is the units digit of 7^13? (a) 9 (b) 3 (c) 1 (d) 7
3. 3.What is the smallest number you must add to 61 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the smallest number you must add to 415 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 4
5. 5.What is the remainder when 172 is divided by 3? (a) 2 (b) 1 (c) 0
6. 6.What is the units digit of 9^59? (a) 8 (b) 7 (c) 9 (d) 1
7. 7.What is the remainder when 704 is divided by 12? (a) 8 (b) 9 (c) 7 (d) 4
8. 8.What is the smallest number you must add to 46 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the smallest number you must add to 815 to make it divisible by 11? (a) 1 (b) 9 (c) 10 (d) 8
10. 10.What is the remainder when 176 is divided by 3? (a) 0 (b) 2 (c) 1
11. 11.What is the remainder when 388 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
12. 12.Which of these numbers is divisible by 11? (a) 5517 (b) 5524 (c) 5506 (d) 5511
13. 13.What is the units digit of 4^55? (a) 5 (b) 6 (c) 3 (d) 4
14. 14.Which of these numbers is divisible by 6? (a) 4439 (b) 4423 (c) 4429 (d) 4428
15. 15.Which of these numbers is divisible by 6? (a) 4663 (b) 4672 (c) 4677 (d) 4668
16. 16.What is the smallest number you must add to 61 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
17. 17.What is the smallest number you must add to 425 to make it divisible by 9? (a) 7 (b) 2 (c) 6 (d) 8
18. 18.What is the smallest number you must add to 445 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 7
19. 19.What is the units digit of 8^18? (a) 8 (b) 6 (c) 2 (d) 4
20. 20.What is the smallest number you must add to 508 to make it divisible by 9? (a) 6 (b) 7 (c) 4 (d) 5
21. 21.What is the units digit of 7^14? (a) 7 (b) 3 (c) 9 (d) 1
22. 22.What is the smallest number you must add to 166 to make it divisible by 4? (a) 3 (b) 1 (c) 2
23. 23.What is the remainder when 349 is divided by 11? (a) 9 (b) 8 (c) 7 (d) 3
24. 24.Which of these numbers is divisible by 6? (a) 781 (b) 776 (c) 774 (d) 773
25. 25.What is the smallest number you must add to 113 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
26. 26.What is the smallest number you must add to 146 to make it divisible by 4? (a) 2 (b) 3 (c) 1
27. 27.Which of these numbers is divisible by 4? (a) 1811 (b) 1817 (c) 1814 (d) 1812
28. 28.What is the units digit of 7^41? (a) 1 (b) 7 (c) 9 (d) 3
29. 29.Which of these numbers is divisible by 8? (a) 2745 (b) 2737 (c) 2733 (d) 2736
30. 30.What is the remainder when 76 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
31. 31.What is the smallest number you must add to 270 to make it divisible by 4? (a) 1 (b) 3 (c) 2
32. 32.What is the smallest number you must add to 662 to make it divisible by 11? (a) 8 (b) 9 (c) 10 (d) 2
33. 33.What is the remainder when 217 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 1113 (b) 1120 (c) 1121 (d) 1133
35. 35.Which of these numbers is divisible by 6? (a) 4926 (b) 4927 (c) 4935 (d) 4924
36. 36.What is the units digit of 8^15? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.What is the units digit of 4^47? (a) 3 (b) 4 (c) 5 (d) 6
38. 38.What is the remainder when 194 is divided by 3? (a) 0 (b) 2 (c) 1
39. 39.Which of these numbers is divisible by 4? (a) 2346 (b) 2343 (c) 2349 (d) 2344
40. 40.Which of these numbers is divisible by 9? (a) 6090 (b) 6095 (c) 6083 (d) 6084

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