Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 922 · 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 103 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
2. 2.What is the smallest number you must add to 698 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 5
3. 3.What is the units digit of 2^45? (a) 4 (b) 8 (c) 2 (d) 6
4. 4.What is the smallest number you must add to 200 to make it divisible by 9? (a) 2 (b) 7 (c) 6 (d) 8
5. 5.What is the remainder when 163 is divided by 3? (a) 2 (b) 0 (c) 1
6. 6.What is the units digit of 3^36? (a) 1 (b) 7 (c) 9 (d) 3
7. 7.Which of these numbers is divisible by 3? (a) 1907 (b) 1904 (c) 1909 (d) 1905
8. 8.Which of these numbers is divisible by 9? (a) 7064 (b) 7052 (c) 7047 (d) 7044
9. 9.What is the remainder when 156 is divided by 11? (a) 2 (b) 3 (c) 9 (d) 1
10. 10.Which of these numbers is divisible by 4? (a) 728 (b) 735 (c) 730 (d) 725
11. 11.Which of these numbers is divisible by 6? (a) 3596 (b) 3594 (c) 3601 (d) 3593
12. 12.What is the remainder when 749 is divided by 12? (a) 7 (b) 6 (c) 4 (d) 5
13. 13.What is the units digit of 3^48? (a) 7 (b) 1 (c) 9 (d) 3
14. 14.What is the units digit of 9^20? (a) 0 (b) 1 (c) 2 (d) 9
15. 15.What is the units digit of 2^28? (a) 6 (b) 2 (c) 4 (d) 8
16. 16.What is the units digit of 7^18? (a) 3 (b) 7 (c) 1 (d) 9
17. 17.What is the remainder when 49 is divided by 3? (a) 1 (b) 2 (c) 0
18. 18.What is the remainder when 622 is divided by 12? (a) 9 (b) 2 (c) 11 (d) 10
19. 19.What is the units digit of 2^18? (a) 4 (b) 2 (c) 8 (d) 6
20. 20.What is the units digit of 3^34? (a) 3 (b) 9 (c) 7 (d) 1
21. 21.What is the remainder when 193 is divided by 9? (a) 3 (b) 6 (c) 4 (d) 5
22. 22.What is the remainder when 748 is divided by 10? (a) 8 (b) 2 (c) 7 (d) 9
23. 23.What is the smallest number you must add to 71 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the remainder when 47 is divided by 3? (a) 2 (b) 0 (c) 1
25. 25.Which of these numbers is divisible by 8? (a) 1137 (b) 1126 (c) 1135 (d) 1128
26. 26.What is the remainder when 623 is divided by 11? (a) 4 (b) 8 (c) 6 (d) 7
27. 27.What is the smallest number you must add to 653 to make it divisible by 11? (a) 8 (b) 6 (c) 4 (d) 7
28. 28.What is the smallest number you must add to 64 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the remainder when 77 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
30. 30.What is the units digit of 3^46? (a) 1 (b) 3 (c) 7 (d) 9
31. 31.Which of these numbers is divisible by 8? (a) 4203 (b) 4222 (c) 4210 (d) 4208
32. 32.What is the smallest number you must add to 626 to make it divisible by 10? (a) 5 (b) 4 (c) 3 (d) 6
33. 33.What is the remainder when 386 is divided by 8? (a) 1 (b) 2 (c) 3 (d) 6
34. 34.What is the remainder when 303 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
35. 35.What is the remainder when 506 is divided by 8? (a) 3 (b) 6 (c) 1 (d) 2
36. 36.What is the smallest number you must add to 460 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 6
37. 37.What is the remainder when 837 is divided by 11? (a) 10 (b) 0 (c) 2 (d) 1
38. 38.Which of these numbers is divisible by 11? (a) 5309 (b) 5322 (c) 5294 (d) 5302
39. 39.What is the units digit of 9^34? (a) 2 (b) 1 (c) 0 (d) 9
40. 40.What is the units digit of 7^31? (a) 1 (b) 3 (c) 9 (d) 7

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