Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 320 · 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 9^22? (a) 0 (b) 2 (c) 1 (d) 9
2. 2.What is the smallest number you must add to 229 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
3. 3.Which of these numbers is divisible by 4? (a) 570 (b) 565 (c) 575 (d) 568
4. 4.Which of these numbers is divisible by 11? (a) 9119 (b) 9123 (c) 9113 (d) 9132
5. 5.What is the units digit of 9^30? (a) 0 (b) 9 (c) 1 (d) 2
6. 6.What is the remainder when 533 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
7. 7.What is the units digit of 9^54? (a) 9 (b) 1 (c) 0 (d) 2
8. 8.What is the remainder when 381 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
9. 9.What is the smallest number you must add to 591 to make it divisible by 10? (a) 1 (b) 7 (c) 8 (d) 9
10. 10.What is the remainder when 37 is divided by 3? (a) 0 (b) 2 (c) 1
11. 11.What is the remainder when 75 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
12. 12.What is the remainder when 167 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
13. 13.What is the smallest number you must add to 229 to make it divisible by 10? (a) 2 (b) 3 (c) 1 (d) 9
14. 14.What is the remainder when 415 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
15. 15.What is the remainder when 204 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
16. 16.Which of these numbers is divisible by 3? (a) 965 (b) 970 (c) 968 (d) 966
17. 17.What is the units digit of 3^44? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.What is the smallest number you must add to 341 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 3? (a) 530 (b) 531 (c) 535 (d) 533
20. 20.What is the units digit of 2^23? (a) 6 (b) 2 (c) 4 (d) 8
21. 21.What is the smallest number you must add to 161 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
22. 22.Which of these numbers is divisible by 9? (a) 7186 (b) 7180 (c) 7171 (d) 7173
23. 23.Which of these numbers is divisible by 11? (a) 3246 (b) 3263 (c) 3245 (d) 3243
24. 24.Which of these numbers is divisible by 3? (a) 1571 (b) 1577 (c) 1572 (d) 1574
25. 25.What is the remainder when 76 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 1
26. 26.What is the smallest number you must add to 861 to make it divisible by 12? (a) 9 (b) 2 (c) 4 (d) 3
27. 27.What is the units digit of 8^50? (a) 4 (b) 8 (c) 6 (d) 2
28. 28.Which of these numbers is divisible by 11? (a) 8935 (b) 8943 (c) 8956 (d) 8953
29. 29.Which of these numbers is divisible by 9? (a) 1064 (b) 1079 (c) 1071 (d) 1085
30. 30.What is the remainder when 115 is divided by 3? (a) 1 (b) 0 (c) 2
31. 31.What is the remainder when 110 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
32. 32.What is the remainder when 529 is divided by 10? (a) 1 (b) 8 (c) 9 (d) 7
33. 33.What is the remainder when 211 is divided by 12? (a) 5 (b) 8 (c) 6 (d) 7
34. 34.What is the smallest number you must add to 636 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5
35. 35.What is the remainder when 277 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
36. 36.Which of these numbers is divisible by 4? (a) 2756 (b) 2759 (c) 2761 (d) 2753
37. 37.Which of these numbers is divisible by 8? (a) 5599 (b) 5604 (c) 5600 (d) 5610
38. 38.What is the units digit of 4^28? (a) 7 (b) 5 (c) 6 (d) 4
39. 39.Which of these numbers is divisible by 11? (a) 7397 (b) 7387 (c) 7380 (d) 7381
40. 40.What is the smallest number you must add to 186 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1

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