Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 216 · 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 3? (a) 2661 (b) 2662 (c) 2659 (d) 2665
2. 2.What is the units digit of 8^37? (a) 4 (b) 6 (c) 8 (d) 2
3. 3.Which of these numbers is divisible by 11? (a) 1795 (b) 1782 (c) 1787 (d) 1772
4. 4.Which of these numbers is divisible by 6? (a) 1489 (b) 1480 (c) 1484 (d) 1482
5. 5.What is the remainder when 391 is divided by 10? (a) 9 (b) 0 (c) 1 (d) 2
6. 6.Which of these numbers is divisible by 8? (a) 5326 (b) 5334 (c) 5320 (d) 5314
7. 7.Which of these numbers is divisible by 9? (a) 392 (b) 378 (c) 374 (d) 383
8. 8.What is the remainder when 306 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
9. 9.What is the smallest number you must add to 74 to make it divisible by 3? (a) 2 (b) 1
10. 10.What is the remainder when 377 is divided by 7? (a) 6 (b) 1 (c) 4 (d) 5
11. 11.What is the smallest number you must add to 837 to make it divisible by 12? (a) 4 (b) 2 (c) 3 (d) 9
12. 12.What is the remainder when 145 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
13. 13.What is the remainder when 608 is divided by 12? (a) 8 (b) 9 (c) 4 (d) 7
14. 14.Which of these numbers is divisible by 11? (a) 1212 (b) 1226 (c) 1221 (d) 1239
15. 15.What is the units digit of 7^54? (a) 3 (b) 7 (c) 9 (d) 1
16. 16.What is the units digit of 3^41? (a) 9 (b) 7 (c) 1 (d) 3
17. 17.Which of these numbers is divisible by 8? (a) 1793 (b) 1784 (c) 1786 (d) 1780
18. 18.What is the remainder when 618 is divided by 11? (a) 9 (b) 2 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 9? (a) 5666 (b) 5652 (c) 5660 (d) 5649
20. 20.Which of these numbers is divisible by 6? (a) 4139 (b) 4145 (c) 4134 (d) 4131
21. 21.What is the smallest number you must add to 197 to make it divisible by 11? (a) 3 (b) 1 (c) 10 (d) 2
22. 22.What is the remainder when 465 is divided by 12? (a) 3 (b) 10 (c) 8 (d) 9
23. 23.Which of these numbers is divisible by 6? (a) 1430 (b) 1434 (c) 1438 (d) 1441
24. 24.What is the remainder when 368 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 6
25. 25.What is the remainder when 158 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
26. 26.What is the remainder when 471 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
27. 27.What is the units digit of 2^50? (a) 4 (b) 8 (c) 6 (d) 2
28. 28.Which of these numbers is divisible by 4? (a) 659 (b) 660 (c) 666 (d) 661
29. 29.What is the remainder when 61 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
30. 30.Which of these numbers is divisible by 4? (a) 2006 (b) 2014 (c) 2009 (d) 2008
31. 31.What is the units digit of 7^53? (a) 3 (b) 9 (c) 7 (d) 1
32. 32.What is the units digit of 3^60? (a) 1 (b) 9 (c) 7 (d) 3
33. 33.What is the remainder when 219 is divided by 10? (a) 9 (b) 1 (c) 8 (d) 7
34. 34.What is the smallest number you must add to 38 to make it divisible by 3? (a) 2 (b) 1
35. 35.What is the remainder when 161 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
36. 36.Which of these numbers is divisible by 11? (a) 8491 (b) 8500 (c) 8481 (d) 8472
37. 37.What is the smallest number you must add to 466 to make it divisible by 11? (a) 4 (b) 8 (c) 7 (d) 6
38. 38.What is the smallest number you must add to 73 to make it divisible by 4? (a) 2 (b) 1 (c) 3
39. 39.Which of these numbers is divisible by 4? (a) 1312 (b) 1311 (c) 1315 (d) 1317
40. 40.What is the remainder when 459 is divided by 12? (a) 2 (b) 4 (c) 3 (d) 9

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