Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 400 · 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) 1 (c) 3 (d) 9
2. 2.What is the units digit of 2^29? (a) 6 (b) 2 (c) 4 (d) 8
3. 3.Which of these numbers is divisible by 11? (a) 7074 (b) 7084 (c) 7100 (d) 7087
4. 4.Which of these numbers is divisible by 6? (a) 3026 (b) 3030 (c) 3037 (d) 3032
5. 5.What is the remainder when 230 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
6. 6.What is the smallest number you must add to 471 to make it divisible by 10? (a) 9 (b) 8 (c) 1 (d) 7
7. 7.What is the remainder when 316 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
8. 8.What is the units digit of 8^54? (a) 6 (b) 8 (c) 2 (d) 4
9. 9.What is the units digit of 4^45? (a) 5 (b) 6 (c) 3 (d) 4
10. 10.What is the remainder when 274 is divided by 12? (a) 2 (b) 9 (c) 11 (d) 10
11. 11.What is the units digit of 8^33? (a) 8 (b) 6 (c) 2 (d) 4
12. 12.What is the remainder when 579 is divided by 9? (a) 3 (b) 6 (c) 2 (d) 4
13. 13.What is the smallest number you must add to 469 to make it divisible by 12? (a) 10 (b) 1 (c) 9 (d) 11
14. 14.Which of these numbers is divisible by 11? (a) 929 (b) 940 (c) 918 (d) 924
15. 15.What is the remainder when 160 is divided by 3? (a) 0 (b) 1 (c) 2
16. 16.What is the units digit of 2^41? (a) 2 (b) 6 (c) 4 (d) 8
17. 17.What is the smallest number you must add to 368 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
18. 18.Which of these numbers is divisible by 3? (a) 885 (b) 890 (c) 883 (d) 887
19. 19.What is the remainder when 257 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 7
20. 20.What is the remainder when 374 is divided by 12? (a) 10 (b) 3 (c) 1 (d) 2
21. 21.What is the units digit of 2^28? (a) 4 (b) 8 (c) 6 (d) 2
22. 22.Which of these numbers is divisible by 3? (a) 904 (b) 899 (c) 900 (d) 902
23. 23.What is the smallest number you must add to 507 to make it divisible by 11? (a) 9 (b) 10 (c) 1 (d) 8
24. 24.What is the units digit of 4^14? (a) 5 (b) 4 (c) 6 (d) 7
25. 25.What is the units digit of 7^50? (a) 1 (b) 3 (c) 7 (d) 9
26. 26.Which of these numbers is divisible by 3? (a) 2657 (b) 2654 (c) 2651 (d) 2652
27. 27.What is the smallest number you must add to 712 to make it divisible by 12? (a) 9 (b) 8 (c) 4 (d) 7
28. 28.What is the smallest number you must add to 918 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
29. 29.What is the remainder when 125 is divided by 12? (a) 5 (b) 7 (c) 4 (d) 6
30. 30.What is the smallest number you must add to 214 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
31. 31.What is the units digit of 9^41? (a) 1 (b) 9 (c) 8 (d) 7
32. 32.Which of these numbers is divisible by 11? (a) 9419 (b) 9427 (c) 9436 (d) 9441
33. 33.What is the units digit of 2^40? (a) 2 (b) 6 (c) 8 (d) 4
34. 34.What is the smallest number you must add to 61 to make it divisible by 3? (a) 1 (b) 2
35. 35.What is the smallest number you must add to 512 to make it divisible by 10? (a) 2 (b) 9 (c) 8 (d) 7
36. 36.What is the remainder when 220 is divided by 3? (a) 1 (b) 0 (c) 2
37. 37.Which of these numbers is divisible by 11? (a) 1791 (b) 1811 (c) 1793 (d) 1798
38. 38.What is the units digit of 7^38? (a) 7 (b) 1 (c) 3 (d) 9
39. 39.What is the remainder when 237 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3
40. 40.What is the smallest number you must add to 141 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3

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