Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 435 · 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 84 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
2. 2.Which of these numbers is divisible by 9? (a) 6400 (b) 6391 (c) 6383 (d) 6390
3. 3.Which of these numbers is divisible by 3? (a) 452 (b) 455 (c) 450 (d) 449
4. 4.What is the smallest number you must add to 133 to make it divisible by 4? (a) 3 (b) 1 (c) 2
5. 5.Which of these numbers is divisible by 4? (a) 847 (b) 840 (c) 839 (d) 842
6. 6.What is the units digit of 2^13? (a) 8 (b) 6 (c) 4 (d) 2
7. 7.Which of these numbers is divisible by 8? (a) 1029 (b) 1032 (c) 1035 (d) 1043
8. 8.Which of these numbers is divisible by 6? (a) 1558 (b) 1563 (c) 1560 (d) 1569
9. 9.What is the units digit of 7^27? (a) 9 (b) 1 (c) 7 (d) 3
10. 10.What is the remainder when 161 is divided by 3? (a) 1 (b) 0 (c) 2
11. 11.What is the smallest number you must add to 230 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the units digit of 9^43? (a) 9 (b) 1 (c) 7 (d) 8
13. 13.What is the remainder when 645 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
14. 14.Which of these numbers is divisible by 3? (a) 806 (b) 803 (c) 799 (d) 801
15. 15.What is the remainder when 271 is divided by 10? (a) 2 (b) 9 (c) 0 (d) 1
16. 16.Which of these numbers is divisible by 4? (a) 1412 (b) 1413 (c) 1410 (d) 1418
17. 17.What is the remainder when 687 is divided by 11? (a) 4 (b) 5 (c) 6 (d) 7
18. 18.What is the remainder when 31 is divided by 3? (a) 0 (b) 2 (c) 1
19. 19.What is the units digit of 4^16? (a) 5 (b) 4 (c) 6 (d) 7
20. 20.Which of these numbers is divisible by 11? (a) 5316 (b) 5324 (c) 5325 (d) 5336
21. 21.What is the smallest number you must add to 79 to make it divisible by 4? (a) 3 (b) 2 (c) 1
22. 22.Which of these numbers is divisible by 6? (a) 4249 (b) 4263 (c) 4258 (d) 4254
23. 23.Which of these numbers is divisible by 4? (a) 229 (b) 239 (c) 232 (d) 235
24. 24.What is the smallest number you must add to 106 to make it divisible by 4? (a) 3 (b) 1 (c) 2
25. 25.What is the smallest number you must add to 228 to make it divisible by 10? (a) 8 (b) 2 (c) 3 (d) 1
26. 26.Which of these numbers is divisible by 11? (a) 1582 (b) 1584 (c) 1599 (d) 1588
27. 27.What is the remainder when 663 is divided by 9? (a) 6 (b) 3 (c) 7 (d) 5
28. 28.What is the remainder when 199 is divided by 8? (a) 1 (b) 7 (c) 5 (d) 6
29. 29.What is the units digit of 4^13? (a) 4 (b) 6 (c) 3 (d) 5
30. 30.What is the units digit of 3^11? (a) 3 (b) 7 (c) 9 (d) 1
31. 31.What is the units digit of 9^15? (a) 1 (b) 8 (c) 7 (d) 9
32. 32.What is the units digit of 8^19? (a) 2 (b) 4 (c) 6 (d) 8
33. 33.What is the smallest number you must add to 285 to make it divisible by 4? (a) 3 (b) 2 (c) 1
34. 34.Which of these numbers is divisible by 11? (a) 6130 (b) 6120 (c) 6116 (d) 6106
35. 35.What is the remainder when 447 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
36. 36.Which of these numbers is divisible by 4? (a) 1797 (b) 1805 (c) 1800 (d) 1801
37. 37.What is the units digit of 2^43? (a) 6 (b) 4 (c) 2 (d) 8
38. 38.Which of these numbers is divisible by 11? (a) 1125 (b) 1143 (c) 1146 (d) 1133
39. 39.Which of these numbers is divisible by 3? (a) 2309 (b) 2306 (c) 2307 (d) 2312
40. 40.What is the units digit of 2^15? (a) 8 (b) 6 (c) 4 (d) 2

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