Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 444 · 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 remainder when 467 is divided by 8? (a) 2 (b) 4 (c) 3 (d) 5
2. 2.What is the remainder when 337 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
3. 3.What is the units digit of 9^12? (a) 0 (b) 9 (c) 2 (d) 1
4. 4.What is the units digit of 4^14? (a) 5 (b) 4 (c) 6 (d) 7
5. 5.What is the remainder when 323 is divided by 10? (a) 4 (b) 2 (c) 3 (d) 7
6. 6.What is the smallest number you must add to 382 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
7. 7.What is the smallest number you must add to 447 to make it divisible by 7? (a) 1 (b) 2 (c) 6 (d) 3
8. 8.Which of these numbers is divisible by 11? (a) 3418 (b) 3396 (c) 3401 (d) 3399
9. 9.Which of these numbers is divisible by 3? (a) 1643 (b) 1636 (c) 1638 (d) 1640
10. 10.Which of these numbers is divisible by 4? (a) 3394 (b) 3389 (c) 3388 (d) 3386
11. 11.Which of these numbers is divisible by 4? (a) 2691 (b) 2697 (c) 2695 (d) 2692
12. 12.What is the units digit of 4^49? (a) 6 (b) 5 (c) 4 (d) 3
13. 13.What is the units digit of 3^21? (a) 7 (b) 9 (c) 1 (d) 3
14. 14.Which of these numbers is divisible by 6? (a) 387 (b) 391 (c) 383 (d) 384
15. 15.What is the smallest number you must add to 43 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the units digit of 3^28? (a) 1 (b) 9 (c) 7 (d) 3
17. 17.Which of these numbers is divisible by 4? (a) 2445 (b) 2444 (c) 2443 (d) 2449
18. 18.Which of these numbers is divisible by 3? (a) 646 (b) 645 (c) 650 (d) 644
19. 19.What is the remainder when 407 is divided by 7? (a) 6 (b) 0 (c) 2 (d) 1
20. 20.What is the units digit of 2^46? (a) 4 (b) 8 (c) 6 (d) 2
21. 21.Which of these numbers is divisible by 6? (a) 5342 (b) 5346 (c) 5350 (d) 5353
22. 22.Which of these numbers is divisible by 9? (a) 5749 (b) 5733 (c) 5730 (d) 5736
23. 23.What is the remainder when 478 is divided by 10? (a) 9 (b) 2 (c) 8 (d) 7
24. 24.What is the units digit of 4^23? (a) 4 (b) 6 (c) 5 (d) 3
25. 25.Which of these numbers is divisible by 8? (a) 379 (b) 373 (c) 363 (d) 368
26. 26.What is the smallest number you must add to 346 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
27. 27.Which of these numbers is divisible by 3? (a) 648 (b) 653 (c) 650 (d) 647
28. 28.Which of these numbers is divisible by 11? (a) 7854 (b) 7844 (c) 7855 (d) 7866
29. 29.Which of these numbers is divisible by 4? (a) 652 (b) 655 (c) 657 (d) 649
30. 30.What is the smallest number you must add to 789 to make it divisible by 12? (a) 2 (b) 4 (c) 9 (d) 3
31. 31.Which of these numbers is divisible by 8? (a) 900 (b) 892 (c) 888 (d) 884
32. 32.What is the smallest number you must add to 618 to make it divisible by 11? (a) 10 (b) 9 (c) 2 (d) 8
33. 33.What is the units digit of 7^49? (a) 9 (b) 1 (c) 3 (d) 7
34. 34.What is the remainder when 372 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
35. 35.What is the units digit of 4^54? (a) 6 (b) 5 (c) 7 (d) 4
36. 36.Which of these numbers is divisible by 3? (a) 757 (b) 763 (c) 761 (d) 759
37. 37.What is the units digit of 9^33? (a) 1 (b) 9 (c) 7 (d) 8
38. 38.Which of these numbers is divisible by 3? (a) 2289 (b) 2293 (c) 2291 (d) 2287
39. 39.What is the smallest number you must add to 215 to make it divisible by 12? (a) 3 (b) 2 (c) 11 (d) 1
40. 40.Which of these numbers is divisible by 9? (a) 6771 (b) 6765 (c) 6768 (d) 6782

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