Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 745 · 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 units digit of 2^60? (a) 8 (b) 2 (c) 4 (d) 6
2. 2.Which of these numbers is divisible by 8? (a) 525 (b) 520 (c) 530 (d) 516
3. 3.What is the remainder when 798 is divided by 11? (a) 8 (b) 5 (c) 7 (d) 6
4. 4.What is the units digit of 8^24? (a) 2 (b) 6 (c) 8 (d) 4
5. 5.What is the smallest number you must add to 523 to make it divisible by 8? (a) 5 (b) 6 (c) 3 (d) 4
6. 6.What is the smallest number you must add to 871 to make it divisible by 11? (a) 2 (b) 10 (c) 9 (d) 8
7. 7.What is the remainder when 352 is divided by 12? (a) 3 (b) 4 (c) 8 (d) 5
8. 8.Which of these numbers is divisible by 8? (a) 7108 (b) 7115 (c) 7101 (d) 7104
9. 9.Which of these numbers is divisible by 3? (a) 2699 (b) 2693 (c) 2696 (d) 2694
10. 10.Which of these numbers is divisible by 4? (a) 957 (b) 960 (c) 962 (d) 966
11. 11.What is the remainder when 800 is divided by 12? (a) 4 (b) 9 (c) 7 (d) 8
12. 12.Which of these numbers is divisible by 9? (a) 1764 (b) 1778 (c) 1766 (d) 1760
13. 13.What is the remainder when 631 is divided by 9? (a) 0 (b) 1 (c) 2 (d) 8
14. 14.What is the smallest number you must add to 115 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 3
15. 15.What is the remainder when 35 is divided by 3? (a) 2 (b) 1 (c) 0
16. 16.What is the remainder when 133 is divided by 10? (a) 7 (b) 2 (c) 3 (d) 4
17. 17.What is the units digit of 4^14? (a) 7 (b) 6 (c) 5 (d) 4
18. 18.Which of these numbers is divisible by 11? (a) 2373 (b) 2389 (c) 2382 (d) 2376
19. 19.What is the smallest number you must add to 834 to make it divisible by 11? (a) 3 (b) 2 (c) 1 (d) 9
20. 20.What is the units digit of 9^14? (a) 9 (b) 2 (c) 1 (d) 0
21. 21.What is the remainder when 659 is divided by 9? (a) 2 (b) 1 (c) 7 (d) 3
22. 22.What is the smallest number you must add to 421 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 5
23. 23.Which of these numbers is divisible by 11? (a) 5407 (b) 5412 (c) 5426 (d) 5416
24. 24.What is the remainder when 314 is divided by 6? (a) 1 (b) 2 (c) 4 (d) 3
25. 25.What is the remainder when 543 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 3
26. 26.What is the smallest number you must add to 787 to make it divisible by 11? (a) 6 (b) 7 (c) 5 (d) 4
27. 27.Which of these numbers is divisible by 8? (a) 3303 (b) 3281 (c) 3288 (d) 3290
28. 28.What is the units digit of 4^10? (a) 7 (b) 6 (c) 5 (d) 4
29. 29.What is the remainder when 93 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
30. 30.What is the remainder when 705 is divided by 9? (a) 6 (b) 4 (c) 2 (d) 3
31. 31.Which of these numbers is divisible by 4? (a) 803 (b) 804 (c) 805 (d) 811
32. 32.What is the remainder when 505 is divided by 12? (a) 11 (b) 2 (c) 1 (d) 0
33. 33.What is the remainder when 679 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 8
34. 34.What is the units digit of 3^54? (a) 3 (b) 1 (c) 9 (d) 7
35. 35.Which of these numbers is divisible by 6? (a) 5281 (b) 5280 (c) 5277 (d) 5288
36. 36.What is the remainder when 202 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
37. 37.Which of these numbers is divisible by 6? (a) 1836 (b) 1846 (c) 1831 (d) 1839
38. 38.Which of these numbers is divisible by 11? (a) 6839 (b) 6821 (c) 6845 (d) 6831
39. 39.Which of these numbers is divisible by 6? (a) 2668 (b) 2664 (c) 2661 (d) 2674
40. 40.Which of these numbers is divisible by 3? (a) 925 (b) 927 (c) 929 (d) 932

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