Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 723 · 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 2^22? (a) 2 (b) 6 (c) 8 (d) 4
2. 2.What is the remainder when 808 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
3. 3.What is the units digit of 7^20? (a) 7 (b) 9 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 4? (a) 3028 (b) 3030 (c) 3033 (d) 3025
5. 5.What is the units digit of 8^15? (a) 2 (b) 4 (c) 8 (d) 6
6. 6.Which of these numbers is divisible by 3? (a) 548 (b) 544 (c) 546 (d) 551
7. 7.Which of these numbers is divisible by 9? (a) 4862 (b) 4849 (c) 4851 (d) 4853
8. 8.What is the remainder when 722 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
9. 9.What is the remainder when 119 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
10. 10.Which of these numbers is divisible by 11? (a) 5368 (b) 5386 (c) 5365 (d) 5375
11. 11.What is the units digit of 8^43? (a) 6 (b) 4 (c) 2 (d) 8
12. 12.What is the remainder when 165 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
13. 13.What is the remainder when 89 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
14. 14.Which of these numbers is divisible by 6? (a) 873 (b) 864 (c) 863 (d) 866
15. 15.Which of these numbers is divisible by 8? (a) 4308 (b) 4301 (c) 4313 (d) 4304
16. 16.What is the units digit of 3^15? (a) 1 (b) 9 (c) 3 (d) 7
17. 17.What is the remainder when 65 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
18. 18.What is the units digit of 8^10? (a) 8 (b) 6 (c) 2 (d) 4
19. 19.Which of these numbers is divisible by 11? (a) 8249 (b) 8268 (c) 8258 (d) 8250
20. 20.What is the smallest number you must add to 739 to make it divisible by 12? (a) 4 (b) 5 (c) 7 (d) 6
21. 21.What is the remainder when 287 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
22. 22.What is the smallest number you must add to 398 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
23. 23.Which of these numbers is divisible by 6? (a) 3939 (b) 3951 (c) 3946 (d) 3942
24. 24.What is the smallest number you must add to 473 to make it divisible by 7? (a) 2 (b) 4 (c) 3 (d) 5
25. 25.What is the units digit of 7^43? (a) 3 (b) 9 (c) 1 (d) 7
26. 26.What is the units digit of 3^43? (a) 1 (b) 7 (c) 3 (d) 9
27. 27.Which of these numbers is divisible by 8? (a) 5879 (b) 5861 (c) 5865 (d) 5864
28. 28.What is the smallest number you must add to 586 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 3
29. 29.Which of these numbers is divisible by 4? (a) 2277 (b) 2281 (c) 2273 (d) 2276
30. 30.What is the remainder when 134 is divided by 12? (a) 1 (b) 2 (c) 3 (d) 10
31. 31.What is the units digit of 9^22? (a) 2 (b) 9 (c) 0 (d) 1
32. 32.Which of these numbers is divisible by 8? (a) 1420 (b) 1424 (c) 1429 (d) 1439
33. 33.Which of these numbers is divisible by 4? (a) 3065 (b) 3059 (c) 3061 (d) 3060
34. 34.What is the remainder when 373 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
35. 35.What is the smallest number you must add to 722 to make it divisible by 12? (a) 9 (b) 10 (c) 2 (d) 11
36. 36.What is the smallest number you must add to 288 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
37. 37.What is the remainder when 227 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
38. 38.Which of these numbers is divisible by 11? (a) 9060 (b) 9042 (c) 9047 (d) 9039
39. 39.What is the remainder when 71 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
40. 40.What is the units digit of 9^41? (a) 1 (b) 8 (c) 7 (d) 9

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