Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 556 · 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 9^27? (a) 8 (b) 7 (c) 9 (d) 1
2. 2.Which of these numbers is divisible by 11? (a) 9070 (b) 9054 (c) 9053 (d) 9045
3. 3.What is the units digit of 4^15? (a) 5 (b) 4 (c) 6 (d) 3
4. 4.What is the remainder when 208 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
5. 5.What is the smallest number you must add to 221 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the smallest number you must add to 230 to make it divisible by 7? (a) 2 (b) 3 (c) 6 (d) 1
7. 7.What is the smallest number you must add to 739 to make it divisible by 11? (a) 2 (b) 8 (c) 9 (d) 10
8. 8.What is the units digit of 9^49? (a) 1 (b) 7 (c) 9 (d) 8
9. 9.What is the remainder when 400 is divided by 9? (a) 5 (b) 3 (c) 4 (d) 6
10. 10.What is the units digit of 9^50? (a) 0 (b) 9 (c) 1 (d) 2
11. 11.What is the remainder when 287 is divided by 8? (a) 7 (b) 5 (c) 1 (d) 6
12. 12.Which of these numbers is divisible by 6? (a) 1045 (b) 1051 (c) 1061 (d) 1050
13. 13.What is the remainder when 463 is divided by 12? (a) 8 (b) 5 (c) 7 (d) 6
14. 14.What is the remainder when 585 is divided by 8? (a) 1 (b) 7 (c) 0 (d) 2
15. 15.Which of these numbers is divisible by 4? (a) 3097 (b) 3095 (c) 3090 (d) 3092
16. 16.What is the remainder when 461 is divided by 6? (a) 5 (b) 3 (c) 1 (d) 4
17. 17.What is the remainder when 383 is divided by 10? (a) 4 (b) 3 (c) 7 (d) 2
18. 18.Which of these numbers is divisible by 8? (a) 2186 (b) 2192 (c) 2201 (d) 2194
19. 19.What is the units digit of 8^54? (a) 4 (b) 2 (c) 8 (d) 6
20. 20.What is the smallest number you must add to 286 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6
21. 21.What is the units digit of 3^32? (a) 9 (b) 7 (c) 1 (d) 3
22. 22.What is the remainder when 50 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
23. 23.What is the smallest number you must add to 423 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
24. 24.What is the units digit of 2^14? (a) 6 (b) 8 (c) 2 (d) 4
25. 25.What is the smallest number you must add to 117 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
26. 26.What is the units digit of 7^35? (a) 7 (b) 9 (c) 1 (d) 3
27. 27.What is the smallest number you must add to 203 to make it divisible by 10? (a) 8 (b) 7 (c) 3 (d) 6
28. 28.What is the smallest number you must add to 99 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
29. 29.What is the smallest number you must add to 636 to make it divisible by 9? (a) 4 (b) 2 (c) 6 (d) 3
30. 30.What is the smallest number you must add to 103 to make it divisible by 3? (a) 1 (b) 2
31. 31.What is the remainder when 276 is divided by 7? (a) 2 (b) 5 (c) 3 (d) 4
32. 32.What is the units digit of 2^18? (a) 8 (b) 6 (c) 4 (d) 2
33. 33.What is the smallest number you must add to 411 to make it divisible by 12? (a) 3 (b) 8 (c) 9 (d) 10
34. 34.Which of these numbers is divisible by 11? (a) 8610 (b) 8591 (c) 8595 (d) 8590
35. 35.Which of these numbers is divisible by 3? (a) 1894 (b) 1893 (c) 1897 (d) 1891
36. 36.Which of these numbers is divisible by 3? (a) 977 (b) 974 (c) 975 (d) 980
37. 37.Which of these numbers is divisible by 11? (a) 1890 (b) 1892 (c) 1911 (d) 1897
38. 38.Which of these numbers is divisible by 4? (a) 2984 (b) 2991 (c) 2983 (d) 2985
39. 39.What is the units digit of 9^23? (a) 9 (b) 1 (c) 7 (d) 8
40. 40.Which of these numbers is divisible by 6? (a) 4231 (b) 4237 (c) 4227 (d) 4230

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