Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 554 · 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 9? (a) 7662 (b) 7649 (c) 7650 (d) 7656
2. 2.Which of these numbers is divisible by 6? (a) 5103 (b) 5106 (c) 5116 (d) 5107
3. 3.What is the remainder when 417 is divided by 11? (a) 1 (b) 9 (c) 10 (d) 8
4. 4.What is the units digit of 4^19? (a) 5 (b) 6 (c) 4 (d) 3
5. 5.Which of these numbers is divisible by 3? (a) 532 (b) 528 (c) 530 (d) 526
6. 6.What is the smallest number you must add to 101 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 2
7. 7.What is the remainder when 328 is divided by 12? (a) 8 (b) 4 (c) 3 (d) 5
8. 8.What is the remainder when 42 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
9. 9.What is the smallest number you must add to 309 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 1
10. 10.What is the remainder when 134 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
11. 11.Which of these numbers is divisible by 3? (a) 2174 (b) 2180 (c) 2177 (d) 2175
12. 12.What is the remainder when 506 is divided by 7? (a) 5 (b) 3 (c) 1 (d) 2
13. 13.What is the remainder when 190 is divided by 3? (a) 2 (b) 1 (c) 0
14. 14.What is the remainder when 458 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
15. 15.What is the smallest number you must add to 181 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the remainder when 82 is divided by 8? (a) 1 (b) 6 (c) 2 (d) 3
17. 17.What is the remainder when 803 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3
18. 18.What is the remainder when 55 is divided by 3? (a) 2 (b) 0 (c) 1
19. 19.Which of these numbers is divisible by 11? (a) 1707 (b) 1697 (c) 1694 (d) 1687
20. 20.What is the remainder when 218 is divided by 11? (a) 8 (b) 9 (c) 10 (d) 2
21. 21.What is the units digit of 2^37? (a) 4 (b) 8 (c) 6 (d) 2
22. 22.What is the units digit of 9^17? (a) 1 (b) 7 (c) 8 (d) 9
23. 23.Which of these numbers is divisible by 3? (a) 1690 (b) 1694 (c) 1689 (d) 1687
24. 24.What is the units digit of 9^34? (a) 2 (b) 9 (c) 0 (d) 1
25. 25.What is the remainder when 240 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
26. 26.Which of these numbers is divisible by 11? (a) 7453 (b) 7460 (c) 7458 (d) 7474
27. 27.What is the smallest number you must add to 384 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
28. 28.What is the remainder when 421 is divided by 7? (a) 1 (b) 0 (c) 6 (d) 2
29. 29.Which of these numbers is divisible by 9? (a) 6366 (b) 6372 (c) 6374 (d) 6386
30. 30.What is the remainder when 217 is divided by 6? (a) 0 (b) 2 (c) 1 (d) 5
31. 31.What is the units digit of 3^27? (a) 3 (b) 9 (c) 7 (d) 1
32. 32.What is the remainder when 142 is divided by 3? (a) 0 (b) 2 (c) 1
33. 33.What is the remainder when 278 is divided by 11? (a) 8 (b) 4 (c) 2 (d) 3
34. 34.What is the smallest number you must add to 56 to make it divisible by 3? (a) 2 (b) 1
35. 35.What is the remainder when 255 is divided by 8? (a) 5 (b) 7 (c) 1 (d) 6
36. 36.Which of these numbers is divisible by 4? (a) 525 (b) 535 (c) 528 (d) 531
37. 37.What is the smallest number you must add to 242 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
38. 38.What is the smallest number you must add to 98 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
39. 39.What is the remainder when 154 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
40. 40.What is the smallest number you must add to 124 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2

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