Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 290 · 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.What is the smallest number you must add to 619 to make it divisible by 12? (a) 4 (b) 7 (c) 6 (d) 5
2. 2.What is the smallest number you must add to 221 to make it divisible by 4? (a) 3 (b) 1 (c) 2
3. 3.What is the remainder when 128 is divided by 3? (a) 0 (b) 2 (c) 1
4. 4.What is the smallest number you must add to 302 to make it divisible by 4? (a) 2 (b) 1 (c) 3
5. 5.Which of these numbers is divisible by 4? (a) 2641 (b) 2644 (c) 2647 (d) 2649
6. 6.Which of these numbers is divisible by 3? (a) 2497 (b) 2493 (c) 2491 (d) 2494
7. 7.What is the smallest number you must add to 764 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
8. 8.Which of these numbers is divisible by 6? (a) 463 (b) 477 (c) 468 (d) 469
9. 9.Which of these numbers is divisible by 9? (a) 1224 (b) 1230 (c) 1239 (d) 1223
10. 10.What is the remainder when 674 is divided by 9? (a) 7 (b) 1 (c) 6 (d) 8
11. 11.What is the remainder when 478 is divided by 12? (a) 9 (b) 10 (c) 2 (d) 11
12. 12.What is the smallest number you must add to 438 to make it divisible by 8? (a) 3 (b) 6 (c) 1 (d) 2
13. 13.Which of these numbers is divisible by 11? (a) 6702 (b) 6713 (c) 6710 (d) 6731
14. 14.Which of these numbers is divisible by 8? (a) 4156 (b) 4140 (c) 4144 (d) 4146
15. 15.What is the remainder when 629 is divided by 11? (a) 2 (b) 9 (c) 1 (d) 3
16. 16.What is the smallest number you must add to 395 to make it divisible by 9? (a) 1 (b) 2 (c) 8 (d) 3
17. 17.What is the units digit of 2^27? (a) 2 (b) 8 (c) 6 (d) 4
18. 18.What is the smallest number you must add to 726 to make it divisible by 9? (a) 3 (b) 2 (c) 6 (d) 4
19. 19.What is the remainder when 287 is divided by 12? (a) 10 (b) 1 (c) 9 (d) 11
20. 20.What is the smallest number you must add to 737 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
21. 21.What is the remainder when 499 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 4
22. 22.Which of these numbers is divisible by 4? (a) 2479 (b) 2483 (c) 2480 (d) 2485
23. 23.What is the remainder when 339 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 7
24. 24.What is the smallest number you must add to 538 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
25. 25.What is the remainder when 43 is divided by 3? (a) 0 (b) 1 (c) 2
26. 26.What is the smallest number you must add to 331 to make it divisible by 12? (a) 7 (b) 4 (c) 5 (d) 6
27. 27.Which of these numbers is divisible by 9? (a) 594 (b) 596 (c) 609 (d) 590
28. 28.What is the remainder when 880 is divided by 12? (a) 8 (b) 3 (c) 5 (d) 4
29. 29.What is the remainder when 91 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
30. 30.What is the units digit of 8^43? (a) 8 (b) 4 (c) 2 (d) 6
31. 31.What is the units digit of 2^22? (a) 4 (b) 2 (c) 8 (d) 6
32. 32.What is the units digit of 4^36? (a) 7 (b) 6 (c) 4 (d) 5
33. 33.Which of these numbers is divisible by 9? (a) 6428 (b) 6420 (c) 6426 (d) 6438
34. 34.What is the smallest number you must add to 673 to make it divisible by 10? (a) 3 (b) 7 (c) 8 (d) 6
35. 35.What is the units digit of 9^21? (a) 7 (b) 9 (c) 1 (d) 8
36. 36.Which of these numbers is divisible by 3? (a) 369 (b) 374 (c) 370 (d) 368
37. 37.What is the remainder when 356 is divided by 6? (a) 1 (b) 2 (c) 4 (d) 3
38. 38.What is the units digit of 4^59? (a) 5 (b) 6 (c) 4 (d) 3
39. 39.What is the remainder when 314 is divided by 8? (a) 1 (b) 3 (c) 2 (d) 6
40. 40.What is the units digit of 3^24? (a) 9 (b) 7 (c) 3 (d) 1

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad