Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 324 · 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 11? (a) 8484 (b) 8468 (c) 8470 (d) 8475
2. 2.What is the units digit of 3^27? (a) 7 (b) 9 (c) 1 (d) 3
3. 3.Which of these numbers is divisible by 11? (a) 6927 (b) 6912 (c) 6935 (d) 6919
4. 4.What is the smallest number you must add to 92 to make it divisible by 3? (a) 1 (b) 2
5. 5.What is the smallest number you must add to 439 to make it divisible by 9? (a) 1 (b) 2 (c) 3 (d) 7
6. 6.What is the remainder when 78 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
7. 7.What is the remainder when 212 is divided by 7? (a) 3 (b) 2 (c) 1 (d) 5
8. 8.Which of these numbers is divisible by 6? (a) 1813 (b) 1828 (c) 1819 (d) 1818
9. 9.What is the units digit of 3^37? (a) 3 (b) 7 (c) 1 (d) 9
10. 10.Which of these numbers is divisible by 3? (a) 782 (b) 788 (c) 783 (d) 784
11. 11.What is the remainder when 713 is divided by 9? (a) 7 (b) 1 (c) 2 (d) 3
12. 12.Which of these numbers is divisible by 8? (a) 4728 (b) 4729 (c) 4738 (d) 4722
13. 13.Which of these numbers is divisible by 3? (a) 158 (b) 164 (c) 160 (d) 159
14. 14.Which of these numbers is divisible by 3? (a) 217 (b) 221 (c) 214 (d) 216
15. 15.Which of these numbers is divisible by 4? (a) 3588 (b) 3589 (c) 3593 (d) 3587
16. 16.What is the units digit of 7^56? (a) 1 (b) 9 (c) 3 (d) 7
17. 17.What is the remainder when 598 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 4
18. 18.What is the units digit of 3^17? (a) 1 (b) 9 (c) 3 (d) 7
19. 19.What is the smallest number you must add to 74 to make it divisible by 3? (a) 1 (b) 2
20. 20.What is the smallest number you must add to 141 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
21. 21.Which of these numbers is divisible by 4? (a) 506 (b) 503 (c) 500 (d) 499
22. 22.What is the remainder when 379 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 2
23. 23.What is the remainder when 122 is divided by 8? (a) 1 (b) 6 (c) 3 (d) 2
24. 24.What is the remainder when 241 is divided by 12? (a) 0 (b) 1 (c) 11 (d) 2
25. 25.What is the remainder when 412 is divided by 7? (a) 4 (b) 5 (c) 1 (d) 6
26. 26.Which of these numbers is divisible by 3? (a) 1661 (b) 1666 (c) 1662 (d) 1664
27. 27.What is the remainder when 221 is divided by 3? (a) 2 (b) 0 (c) 1
28. 28.What is the smallest number you must add to 113 to make it divisible by 4? (a) 1 (b) 3 (c) 2
29. 29.Which of these numbers is divisible by 9? (a) 4143 (b) 4137 (c) 4140 (d) 4155
30. 30.What is the smallest number you must add to 69 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
31. 31.What is the units digit of 4^16? (a) 4 (b) 5 (c) 7 (d) 6
32. 32.What is the units digit of 8^59? (a) 8 (b) 4 (c) 6 (d) 2
33. 33.What is the remainder when 364 is divided by 11? (a) 10 (b) 2 (c) 0 (d) 1
34. 34.What is the smallest number you must add to 256 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
35. 35.What is the smallest number you must add to 227 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
36. 36.What is the smallest number you must add to 333 to make it divisible by 12? (a) 4 (b) 3 (c) 9 (d) 2
37. 37.What is the smallest number you must add to 396 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
38. 38.What is the remainder when 94 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
39. 39.What is the remainder when 159 is divided by 12? (a) 3 (b) 2 (c) 4 (d) 9
40. 40.What is the smallest number you must add to 381 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5

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