Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 270 · 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 729 is divided by 12? (a) 10 (b) 8 (c) 9 (d) 3
2. 2.What is the units digit of 4^11? (a) 4 (b) 3 (c) 5 (d) 6
3. 3.What is the units digit of 4^59? (a) 3 (b) 5 (c) 4 (d) 6
4. 4.Which of these numbers is divisible by 9? (a) 2790 (b) 2794 (c) 2787 (d) 2806
5. 5.What is the units digit of 4^41? (a) 4 (b) 5 (c) 3 (d) 6
6. 6.What is the units digit of 8^19? (a) 6 (b) 2 (c) 4 (d) 8
7. 7.Which of these numbers is divisible by 11? (a) 3953 (b) 3933 (c) 3947 (d) 3938
8. 8.What is the remainder when 393 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
9. 9.What is the smallest number you must add to 124 to make it divisible by 6? (a) 4 (b) 1 (c) 2 (d) 3
10. 10.What is the units digit of 9^30? (a) 9 (b) 2 (c) 0 (d) 1
11. 11.What is the remainder when 534 is divided by 12? (a) 8 (b) 6 (c) 7 (d) 5
12. 12.What is the remainder when 194 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
13. 13.What is the remainder when 126 is divided by 8? (a) 2 (b) 7 (c) 5 (d) 6
14. 14.Which of these numbers is divisible by 9? (a) 1143 (b) 1150 (c) 1135 (d) 1154
15. 15.What is the smallest number you must add to 269 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 3
16. 16.What is the units digit of 4^34? (a) 7 (b) 6 (c) 5 (d) 4
17. 17.Which of these numbers is divisible by 8? (a) 3698 (b) 3694 (c) 3696 (d) 3706
18. 18.What is the smallest number you must add to 687 to make it divisible by 11? (a) 5 (b) 6 (c) 8 (d) 7
19. 19.What is the remainder when 573 is divided by 9? (a) 7 (b) 3 (c) 5 (d) 6
20. 20.What is the units digit of 2^51? (a) 6 (b) 2 (c) 4 (d) 8
21. 21.What is the remainder when 383 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
22. 22.What is the units digit of 2^43? (a) 2 (b) 6 (c) 4 (d) 8
23. 23.What is the remainder when 215 is divided by 6? (a) 5 (b) 4 (c) 1 (d) 3
24. 24.What is the smallest number you must add to 346 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
25. 25.What is the smallest number you must add to 270 to make it divisible by 12? (a) 7 (b) 8 (c) 6 (d) 5
26. 26.What is the units digit of 4^12? (a) 4 (b) 7 (c) 6 (d) 5
27. 27.What is the units digit of 3^11? (a) 3 (b) 9 (c) 7 (d) 1
28. 28.Which of these numbers is divisible by 8? (a) 6095 (b) 6098 (c) 6110 (d) 6096
29. 29.What is the units digit of 8^27? (a) 8 (b) 6 (c) 4 (d) 2
30. 30.What is the smallest number you must add to 427 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
31. 31.What is the smallest number you must add to 280 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
32. 32.Which of these numbers is divisible by 4? (a) 3590 (b) 3586 (c) 3584 (d) 3581
33. 33.What is the units digit of 7^32? (a) 7 (b) 9 (c) 1 (d) 3
34. 34.Which of these numbers is divisible by 6? (a) 1645 (b) 1642 (c) 1651 (d) 1644
35. 35.What is the units digit of 3^26? (a) 3 (b) 7 (c) 1 (d) 9
36. 36.Which of these numbers is divisible by 11? (a) 8283 (b) 8300 (c) 8293 (d) 8278
37. 37.What is the remainder when 267 is divided by 10? (a) 3 (b) 8 (c) 6 (d) 7
38. 38.What is the remainder when 413 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
39. 39.What is the units digit of 4^29? (a) 6 (b) 3 (c) 5 (d) 4
40. 40.What is the remainder when 371 is divided by 12? (a) 11 (b) 9 (c) 10 (d) 1

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