Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 833 · 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 4? (a) 190 (b) 188 (c) 195 (d) 185
2. 2.What is the smallest number you must add to 502 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9
3. 3.What is the smallest number you must add to 894 to make it divisible by 12? (a) 5 (b) 8 (c) 6 (d) 7
4. 4.Which of these numbers is divisible by 8? (a) 4089 (b) 4110 (c) 4102 (d) 4096
5. 5.Which of these numbers is divisible by 9? (a) 614 (b) 603 (c) 602 (d) 609
6. 6.Which of these numbers is divisible by 6? (a) 2725 (b) 2716 (c) 2718 (d) 2721
7. 7.What is the remainder when 707 is divided by 10? (a) 8 (b) 6 (c) 3 (d) 7
8. 8.What is the smallest number you must add to 217 to make it divisible by 9? (a) 7 (b) 6 (c) 1 (d) 8
9. 9.Which of these numbers is divisible by 3? (a) 491 (b) 493 (c) 492 (d) 496
10. 10.Which of these numbers is divisible by 11? (a) 6459 (b) 6454 (c) 6470 (d) 6457
11. 11.What is the units digit of 2^11? (a) 2 (b) 8 (c) 4 (d) 6
12. 12.What is the smallest number you must add to 521 to make it divisible by 11? (a) 4 (b) 7 (c) 6 (d) 8
13. 13.Which of these numbers is divisible by 4? (a) 2002 (b) 2000 (c) 2005 (d) 1998
14. 14.Which of these numbers is divisible by 4? (a) 720 (b) 717 (c) 727 (d) 723
15. 15.What is the remainder when 115 is divided by 6? (a) 0 (b) 2 (c) 5 (d) 1
16. 16.What is the smallest number you must add to 170 to make it divisible by 3? (a) 2 (b) 1
17. 17.Which of these numbers is divisible by 9? (a) 6654 (b) 6642 (c) 6640 (d) 6643
18. 18.What is the remainder when 80 is divided by 3? (a) 0 (b) 2 (c) 1
19. 19.What is the units digit of 3^12? (a) 7 (b) 9 (c) 3 (d) 1
20. 20.What is the units digit of 8^35? (a) 2 (b) 8 (c) 4 (d) 6
21. 21.What is the units digit of 7^38? (a) 7 (b) 1 (c) 9 (d) 3
22. 22.What is the smallest number you must add to 542 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
23. 23.What is the smallest number you must add to 172 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
24. 24.Which of these numbers is divisible by 9? (a) 2331 (b) 2338 (c) 2345 (d) 2325
25. 25.What is the smallest number you must add to 271 to make it divisible by 4? (a) 2 (b) 3 (c) 1
26. 26.What is the units digit of 7^56? (a) 7 (b) 1 (c) 3 (d) 9
27. 27.Which of these numbers is divisible by 8? (a) 790 (b) 782 (c) 794 (d) 784
28. 28.Which of these numbers is divisible by 9? (a) 1193 (b) 1185 (c) 1179 (d) 1171
29. 29.What is the units digit of 9^13? (a) 8 (b) 7 (c) 9 (d) 1
30. 30.What is the remainder when 509 is divided by 12? (a) 7 (b) 4 (c) 5 (d) 6
31. 31.What is the remainder when 63 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
32. 32.Which of these numbers is divisible by 8? (a) 1503 (b) 1504 (c) 1510 (d) 1515
33. 33.What is the remainder when 208 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
34. 34.What is the units digit of 2^10? (a) 6 (b) 4 (c) 8 (d) 2
35. 35.What is the smallest number you must add to 188 to make it divisible by 9? (a) 1 (b) 2 (c) 8 (d) 3
36. 36.What is the smallest number you must add to 209 to make it divisible by 3? (a) 2 (b) 1
37. 37.Which of these numbers is divisible by 6? (a) 3551 (b) 3546 (c) 3556 (d) 3544
38. 38.What is the remainder when 464 is divided by 12? (a) 4 (b) 9 (c) 8 (d) 7
39. 39.What is the smallest number you must add to 374 to make it divisible by 12? (a) 10 (b) 2 (c) 9 (d) 11
40. 40.What is the units digit of 7^31? (a) 1 (b) 9 (c) 3 (d) 7

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