Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 853 · 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 6? (a) 4174 (b) 4177 (c) 4176 (d) 4185
2. 2.What is the remainder when 83 is divided by 3? (a) 0 (b) 1 (c) 2
3. 3.Which of these numbers is divisible by 8? (a) 629 (b) 618 (c) 624 (d) 637
4. 4.Which of these numbers is divisible by 8? (a) 1380 (b) 1372 (c) 1376 (d) 1391
5. 5.What is the remainder when 79 is divided by 3? (a) 0 (b) 1 (c) 2
6. 6.What is the remainder when 457 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
7. 7.What is the units digit of 9^19? (a) 1 (b) 8 (c) 9 (d) 7
8. 8.What is the units digit of 9^32? (a) 9 (b) 1 (c) 0 (d) 2
9. 9.What is the units digit of 4^17? (a) 3 (b) 4 (c) 5 (d) 6
10. 10.What is the smallest number you must add to 118 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the units digit of 7^22? (a) 3 (b) 1 (c) 7 (d) 9
12. 12.What is the remainder when 567 is divided by 12? (a) 2 (b) 4 (c) 3 (d) 9
13. 13.What is the remainder when 638 is divided by 9? (a) 1 (b) 8 (c) 7 (d) 6
14. 14.What is the units digit of 4^19? (a) 4 (b) 6 (c) 3 (d) 5
15. 15.What is the remainder when 376 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
16. 16.What is the remainder when 289 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
17. 17.What is the remainder when 122 is divided by 6? (a) 2 (b) 1 (c) 3 (d) 4
18. 18.What is the units digit of 8^25? (a) 4 (b) 8 (c) 6 (d) 2
19. 19.What is the units digit of 9^59? (a) 9 (b) 1 (c) 7 (d) 8
20. 20.What is the remainder when 163 is divided by 3? (a) 2 (b) 0 (c) 1
21. 21.What is the remainder when 165 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
22. 22.What is the smallest number you must add to 166 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
23. 23.Which of these numbers is divisible by 9? (a) 1444 (b) 1449 (c) 1457 (d) 1463
24. 24.What is the smallest number you must add to 153 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
25. 25.What is the smallest number you must add to 137 to make it divisible by 11? (a) 8 (b) 6 (c) 5 (d) 7
26. 26.What is the smallest number you must add to 358 to make it divisible by 6? (a) 1 (b) 4 (c) 2 (d) 3
27. 27.What is the remainder when 55 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
28. 28.What is the smallest number you must add to 141 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
29. 29.What is the smallest number you must add to 403 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
30. 30.What is the units digit of 3^43? (a) 7 (b) 1 (c) 9 (d) 3
31. 31.What is the smallest number you must add to 445 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 3
32. 32.What is the remainder when 265 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
33. 33.What is the units digit of 8^37? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.Which of these numbers is divisible by 4? (a) 539 (b) 530 (c) 534 (d) 532
35. 35.What is the smallest number you must add to 237 to make it divisible by 7? (a) 2 (b) 6 (c) 3 (d) 1
36. 36.What is the units digit of 4^24? (a) 6 (b) 5 (c) 4 (d) 7
37. 37.What is the units digit of 4^41? (a) 3 (b) 4 (c) 6 (d) 5
38. 38.Which of these numbers is divisible by 11? (a) 6083 (b) 6081 (c) 6087 (d) 6104
39. 39.What is the units digit of 9^13? (a) 9 (b) 7 (c) 1 (d) 8
40. 40.Which of these numbers is divisible by 8? (a) 5536 (b) 5542 (c) 5530 (d) 5547

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