Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 276 · 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) 3790 (b) 3811 (c) 3798 (d) 3805
2. 2.What is the remainder when 830 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 4
3. 3.What is the remainder when 264 is divided by 10? (a) 3 (b) 4 (c) 6 (d) 5
4. 4.What is the remainder when 674 is divided by 9? (a) 1 (b) 6 (c) 7 (d) 8
5. 5.What is the remainder when 95 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 5
6. 6.What is the units digit of 4^12? (a) 4 (b) 5 (c) 7 (d) 6
7. 7.What is the remainder when 441 is divided by 8? (a) 7 (b) 0 (c) 1 (d) 2
8. 8.What is the units digit of 7^17? (a) 7 (b) 3 (c) 1 (d) 9
9. 9.What is the units digit of 3^57? (a) 3 (b) 1 (c) 7 (d) 9
10. 10.Which of these numbers is divisible by 11? (a) 3982 (b) 3978 (c) 3995 (d) 3991
11. 11.What is the units digit of 2^30? (a) 6 (b) 2 (c) 8 (d) 4
12. 12.Which of these numbers is divisible by 3? (a) 1362 (b) 1366 (c) 1364 (d) 1361
13. 13.Which of these numbers is divisible by 6? (a) 812 (b) 804 (c) 799 (d) 807
14. 14.What is the remainder when 149 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
15. 15.Which of these numbers is divisible by 9? (a) 4134 (b) 4143 (c) 4131 (d) 4123
16. 16.Which of these numbers is divisible by 3? (a) 1414 (b) 1412 (c) 1413 (d) 1417
17. 17.What is the units digit of 2^54? (a) 2 (b) 4 (c) 8 (d) 6
18. 18.What is the remainder when 147 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
19. 19.Which of these numbers is divisible by 11? (a) 8365 (b) 8349 (c) 8342 (d) 8354
20. 20.What is the smallest number you must add to 512 to make it divisible by 10? (a) 8 (b) 7 (c) 2 (d) 9
21. 21.What is the units digit of 8^14? (a) 2 (b) 8 (c) 6 (d) 4
22. 22.Which of these numbers is divisible by 11? (a) 5343 (b) 5353 (c) 5360 (d) 5346
23. 23.Which of these numbers is divisible by 4? (a) 979 (b) 975 (c) 981 (d) 976
24. 24.Which of these numbers is divisible by 6? (a) 5146 (b) 5153 (c) 5159 (d) 5148
25. 25.What is the remainder when 121 is divided by 3? (a) 2 (b) 0 (c) 1
26. 26.What is the units digit of 3^30? (a) 7 (b) 1 (c) 9 (d) 3
27. 27.Which of these numbers is divisible by 6? (a) 3475 (b) 3463 (c) 3471 (d) 3468
28. 28.What is the remainder when 266 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
29. 29.Which of these numbers is divisible by 9? (a) 7824 (b) 7807 (c) 7818 (d) 7812
30. 30.What is the remainder when 213 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
31. 31.What is the remainder when 236 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5
32. 32.Which of these numbers is divisible by 4? (a) 249 (b) 253 (c) 246 (d) 248
33. 33.What is the smallest number you must add to 456 to make it divisible by 10? (a) 6 (b) 4 (c) 3 (d) 5
34. 34.Which of these numbers is divisible by 8? (a) 6890 (b) 6906 (c) 6898 (d) 6896
35. 35.What is the smallest number you must add to 442 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 2
36. 36.What is the units digit of 8^15? (a) 2 (b) 6 (c) 8 (d) 4
37. 37.What is the units digit of 9^19? (a) 8 (b) 9 (c) 1 (d) 7
38. 38.What is the units digit of 8^57? (a) 4 (b) 6 (c) 8 (d) 2
39. 39.What is the smallest number you must add to 397 to make it divisible by 9? (a) 7 (b) 6 (c) 1 (d) 8
40. 40.What is the smallest number you must add to 146 to make it divisible by 3? (a) 2 (b) 1

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