Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 188 · 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 7^55? (a) 7 (b) 3 (c) 1 (d) 9
2. 2.What is the remainder when 252 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
3. 3.What is the remainder when 173 is divided by 11? (a) 8 (b) 3 (c) 7 (d) 9
4. 4.What is the units digit of 3^11? (a) 3 (b) 1 (c) 7 (d) 9
5. 5.What is the units digit of 8^36? (a) 8 (b) 2 (c) 4 (d) 6
6. 6.What is the remainder when 74 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
7. 7.Which of these numbers is divisible by 6? (a) 1465 (b) 1461 (c) 1471 (d) 1464
8. 8.What is the remainder when 277 is divided by 10? (a) 3 (b) 8 (c) 7 (d) 6
9. 9.What is the units digit of 3^31? (a) 1 (b) 7 (c) 9 (d) 3
10. 10.Which of these numbers is divisible by 8? (a) 2438 (b) 2451 (c) 2447 (d) 2440
11. 11.Which of these numbers is divisible by 8? (a) 1424 (b) 1434 (c) 1418 (d) 1431
12. 12.What is the units digit of 3^15? (a) 7 (b) 1 (c) 3 (d) 9
13. 13.What is the smallest number you must add to 453 to make it divisible by 12? (a) 9 (b) 2 (c) 3 (d) 4
14. 14.What is the smallest number you must add to 111 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 624 (b) 625 (c) 622 (d) 629
16. 16.Which of these numbers is divisible by 3? (a) 1142 (b) 1144 (c) 1140 (d) 1138
17. 17.What is the units digit of 7^33? (a) 3 (b) 1 (c) 9 (d) 7
18. 18.Which of these numbers is divisible by 11? (a) 6740 (b) 6732 (c) 6723 (d) 6750
19. 19.What is the units digit of 9^17? (a) 7 (b) 8 (c) 1 (d) 9
20. 20.What is the remainder when 621 is divided by 10? (a) 2 (b) 9 (c) 0 (d) 1
21. 21.What is the smallest number you must add to 140 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
22. 22.What is the remainder when 615 is divided by 12? (a) 4 (b) 3 (c) 9 (d) 2
23. 23.Which of these numbers is divisible by 11? (a) 917 (b) 904 (c) 902 (d) 897
24. 24.Which of these numbers is divisible by 3? (a) 785 (b) 781 (c) 788 (d) 783
25. 25.Which of these numbers is divisible by 11? (a) 2456 (b) 2451 (c) 2432 (d) 2442
26. 26.What is the remainder when 776 is divided by 11? (a) 7 (b) 6 (c) 8 (d) 5
27. 27.What is the units digit of 9^47? (a) 8 (b) 7 (c) 9 (d) 1
28. 28.Which of these numbers is divisible by 3? (a) 2556 (b) 2561 (c) 2557 (d) 2554
29. 29.Which of these numbers is divisible by 6? (a) 3140 (b) 3132 (c) 3128 (d) 3136
30. 30.Which of these numbers is divisible by 11? (a) 1115 (b) 1104 (c) 1123 (d) 1111
31. 31.What is the smallest number you must add to 109 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
32. 32.What is the units digit of 4^19? (a) 6 (b) 3 (c) 4 (d) 5
33. 33.What is the remainder when 470 is divided by 9? (a) 7 (b) 1 (c) 3 (d) 2
34. 34.What is the smallest number you must add to 591 to make it divisible by 11? (a) 3 (b) 2 (c) 4 (d) 8
35. 35.What is the remainder when 165 is divided by 7? (a) 6 (b) 5 (c) 3 (d) 4
36. 36.What is the units digit of 8^26? (a) 6 (b) 8 (c) 2 (d) 4
37. 37.What is the units digit of 3^14? (a) 7 (b) 9 (c) 1 (d) 3
38. 38.What is the smallest number you must add to 791 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 1
39. 39.What is the smallest number you must add to 114 to make it divisible by 8? (a) 5 (b) 7 (c) 6 (d) 2
40. 40.What is the smallest number you must add to 747 to make it divisible by 12? (a) 10 (b) 9 (c) 8 (d) 3

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