Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 524 · 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.Which of these numbers is divisible by 9? (a) 3867 (b) 3859 (c) 3849 (d) 3852
2. 2.Which of these numbers is divisible by 3? (a) 397 (b) 396 (c) 401 (d) 394
3. 3.Which of these numbers is divisible by 8? (a) 1695 (b) 1700 (c) 1682 (d) 1688
4. 4.What is the remainder when 169 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
5. 5.What is the units digit of 3^38? (a) 1 (b) 9 (c) 7 (d) 3
6. 6.What is the smallest number you must add to 274 to make it divisible by 4? (a) 1 (b) 3 (c) 2
7. 7.Which of these numbers is divisible by 11? (a) 7911 (b) 7923 (c) 7905 (d) 7909
8. 8.What is the units digit of 3^13? (a) 3 (b) 1 (c) 7 (d) 9
9. 9.Which of these numbers is divisible by 9? (a) 5120 (b) 5097 (c) 5103 (d) 5106
10. 10.Which of these numbers is divisible by 11? (a) 3630 (b) 3636 (c) 3642 (d) 3625
11. 11.What is the remainder when 169 is divided by 3? (a) 2 (b) 0 (c) 1
12. 12.Which of these numbers is divisible by 6? (a) 1195 (b) 1193 (c) 1194 (d) 1201
13. 13.What is the remainder when 490 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 997 (b) 995 (c) 991 (d) 993
15. 15.What is the units digit of 8^29? (a) 6 (b) 4 (c) 2 (d) 8
16. 16.What is the smallest number you must add to 486 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
17. 17.Which of these numbers is divisible by 8? (a) 6717 (b) 6712 (c) 6710 (d) 6723
18. 18.What is the units digit of 8^56? (a) 8 (b) 2 (c) 4 (d) 6
19. 19.Which of these numbers is divisible by 8? (a) 1050 (b) 1039 (c) 1042 (d) 1040
20. 20.What is the smallest number you must add to 338 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
21. 21.Which of these numbers is divisible by 9? (a) 5193 (b) 5192 (c) 5200 (d) 5207
22. 22.Which of these numbers is divisible by 8? (a) 5059 (b) 5070 (c) 5052 (d) 5056
23. 23.Which of these numbers is divisible by 8? (a) 3270 (b) 3275 (c) 3264 (d) 3259
24. 24.What is the remainder when 115 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 4
25. 25.What is the smallest number you must add to 67 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
26. 26.Which of these numbers is divisible by 8? (a) 6888 (b) 6885 (c) 6897 (d) 6891
27. 27.Which of these numbers is divisible by 9? (a) 4883 (b) 4899 (c) 4887 (d) 4888
28. 28.Which of these numbers is divisible by 8? (a) 6870 (b) 6872 (c) 6878 (d) 6883
29. 29.What is the smallest number you must add to 510 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
30. 30.What is the units digit of 8^13? (a) 6 (b) 2 (c) 8 (d) 4
31. 31.What is the remainder when 444 is divided by 8? (a) 4 (b) 5 (c) 6 (d) 3
32. 32.What is the smallest number you must add to 342 to make it divisible by 10? (a) 9 (b) 8 (c) 7 (d) 2
33. 33.What is the smallest number you must add to 542 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 5485 (b) 5480 (c) 5477 (d) 5493
35. 35.What is the remainder when 80 is divided by 6? (a) 3 (b) 1 (c) 2 (d) 4
36. 36.Which of these numbers is divisible by 11? (a) 2596 (b) 2597 (c) 2589 (d) 2614
37. 37.What is the remainder when 139 is divided by 12? (a) 5 (b) 8 (c) 7 (d) 6
38. 38.What is the remainder when 669 is divided by 11? (a) 10 (b) 9 (c) 8 (d) 2
39. 39.What is the units digit of 3^48? (a) 1 (b) 3 (c) 9 (d) 7
40. 40.What is the units digit of 9^34? (a) 1 (b) 0 (c) 2 (d) 9

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