Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 610 · 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 501 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 2
2. 2.Which of these numbers is divisible by 3? (a) 1231 (b) 1235 (c) 1228 (d) 1230
3. 3.What is the smallest number you must add to 207 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
4. 4.Which of these numbers is divisible by 9? (a) 5319 (b) 5321 (c) 5329 (d) 5316
5. 5.What is the units digit of 8^31? (a) 6 (b) 8 (c) 2 (d) 4
6. 6.Which of these numbers is divisible by 3? (a) 420 (b) 422 (c) 418 (d) 424
7. 7.What is the remainder when 200 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 6
8. 8.What is the units digit of 9^25? (a) 9 (b) 7 (c) 1 (d) 8
9. 9.What is the units digit of 2^12? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.Which of these numbers is divisible by 3? (a) 2119 (b) 2113 (c) 2116 (d) 2115
11. 11.What is the units digit of 9^47? (a) 1 (b) 9 (c) 8 (d) 7
12. 12.What is the smallest number you must add to 349 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3
13. 13.What is the units digit of 8^17? (a) 4 (b) 2 (c) 6 (d) 8
14. 14.What is the remainder when 279 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
15. 15.Which of these numbers is divisible by 4? (a) 676 (b) 679 (c) 682 (d) 673
16. 16.What is the remainder when 559 is divided by 9? (a) 0 (b) 8 (c) 2 (d) 1
17. 17.What is the remainder when 156 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 1
18. 18.What is the remainder when 373 is divided by 11? (a) 10 (b) 9 (c) 1 (d) 8
19. 19.Which of these numbers is divisible by 4? (a) 691 (b) 684 (c) 685 (d) 683
20. 20.What is the smallest number you must add to 674 to make it divisible by 11? (a) 9 (b) 7 (c) 3 (d) 8
21. 21.What is the smallest number you must add to 73 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
22. 22.Which of these numbers is divisible by 4? (a) 3044 (b) 3049 (c) 3047 (d) 3041
23. 23.Which of these numbers is divisible by 6? (a) 1907 (b) 1908 (c) 1911 (d) 1918
24. 24.What is the units digit of 3^21? (a) 7 (b) 3 (c) 9 (d) 1
25. 25.Which of these numbers is divisible by 3? (a) 475 (b) 482 (c) 477 (d) 478
26. 26.What is the remainder when 343 is divided by 12? (a) 8 (b) 6 (c) 5 (d) 7
27. 27.What is the smallest number you must add to 600 to make it divisible by 9? (a) 2 (b) 6 (c) 3 (d) 4
28. 28.What is the units digit of 7^60? (a) 7 (b) 1 (c) 3 (d) 9
29. 29.What is the units digit of 4^30? (a) 6 (b) 4 (c) 5 (d) 7
30. 30.What is the units digit of 2^58? (a) 6 (b) 4 (c) 2 (d) 8
31. 31.What is the smallest number you must add to 713 to make it divisible by 10? (a) 3 (b) 7 (c) 6 (d) 8
32. 32.Which of these numbers is divisible by 8? (a) 3129 (b) 3119 (c) 3120 (d) 3122
33. 33.Which of these numbers is divisible by 9? (a) 5723 (b) 5705 (c) 5709 (d) 5706
34. 34.What is the remainder when 897 is divided by 12? (a) 10 (b) 3 (c) 8 (d) 9
35. 35.Which of these numbers is divisible by 8? (a) 5942 (b) 5928 (c) 5931 (d) 5923
36. 36.Which of these numbers is divisible by 8? (a) 5693 (b) 5680 (c) 5673 (d) 5682
37. 37.What is the units digit of 7^21? (a) 7 (b) 3 (c) 9 (d) 1
38. 38.Which of these numbers is divisible by 3? (a) 1677 (b) 1678 (c) 1676 (d) 1682
39. 39.What is the remainder when 197 is divided by 3? (a) 2 (b) 1 (c) 0
40. 40.Which of these numbers is divisible by 8? (a) 5391 (b) 5392 (c) 5399 (d) 5401

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