Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 918 · 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 7^32? (a) 9 (b) 1 (c) 3 (d) 7
2. 2.What is the units digit of 9^32? (a) 2 (b) 0 (c) 1 (d) 9
3. 3.What is the smallest number you must add to 588 to make it divisible by 10? (a) 1 (b) 2 (c) 8 (d) 3
4. 4.Which of these numbers is divisible by 3? (a) 1892 (b) 1889 (c) 1887 (d) 1886
5. 5.Which of these numbers is divisible by 3? (a) 973 (b) 970 (c) 969 (d) 967
6. 6.What is the remainder when 156 is divided by 11? (a) 9 (b) 3 (c) 1 (d) 2
7. 7.Which of these numbers is divisible by 3? (a) 373 (b) 377 (c) 380 (d) 375
8. 8.Which of these numbers is divisible by 4? (a) 2799 (b) 2805 (c) 2800 (d) 2803
9. 9.What is the smallest number you must add to 296 to make it divisible by 10? (a) 6 (b) 4 (c) 3 (d) 5
10. 10.What is the remainder when 533 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
11. 11.Which of these numbers is divisible by 9? (a) 2578 (b) 2584 (c) 2600 (d) 2583
12. 12.What is the units digit of 4^38? (a) 7 (b) 4 (c) 5 (d) 6
13. 13.What is the remainder when 74 is divided by 3? (a) 2 (b) 0 (c) 1
14. 14.What is the smallest number you must add to 347 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4
15. 15.What is the units digit of 8^50? (a) 4 (b) 2 (c) 8 (d) 6
16. 16.What is the remainder when 253 is divided by 6? (a) 1 (b) 0 (c) 5 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 463 (b) 458 (c) 460 (d) 465
18. 18.What is the smallest number you must add to 752 to make it divisible by 12? (a) 5 (b) 4 (c) 8 (d) 3
19. 19.What is the remainder when 698 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
20. 20.What is the smallest number you must add to 642 to make it divisible by 12? (a) 5 (b) 8 (c) 7 (d) 6
21. 21.What is the remainder when 863 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 4
22. 22.What is the units digit of 8^42? (a) 4 (b) 6 (c) 2 (d) 8
23. 23.What is the smallest number you must add to 204 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 5
24. 24.What is the remainder when 643 is divided by 12? (a) 6 (b) 8 (c) 7 (d) 5
25. 25.Which of these numbers is divisible by 8? (a) 1888 (b) 1882 (c) 1898 (d) 1895
26. 26.What is the remainder when 178 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
27. 27.What is the remainder when 181 is divided by 3? (a) 1 (b) 2 (c) 0
28. 28.Which of these numbers is divisible by 6? (a) 4057 (b) 4052 (c) 4050 (d) 4048
29. 29.What is the smallest number you must add to 461 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 2
30. 30.Which of these numbers is divisible by 8? (a) 6616 (b) 6610 (c) 6626 (d) 6621
31. 31.What is the units digit of 9^13? (a) 7 (b) 9 (c) 8 (d) 1
32. 32.What is the smallest number you must add to 170 to make it divisible by 9? (a) 2 (b) 1 (c) 3 (d) 8
33. 33.What is the smallest number you must add to 205 to make it divisible by 3? (a) 1 (b) 2
34. 34.Which of these numbers is divisible by 6? (a) 4321 (b) 4317 (c) 4329 (d) 4320
35. 35.Which of these numbers is divisible by 9? (a) 7821 (b) 7825 (c) 7818 (d) 7832
36. 36.Which of these numbers is divisible by 11? (a) 6274 (b) 6260 (c) 6288 (d) 6270
37. 37.Which of these numbers is divisible by 11? (a) 1892 (b) 1885 (c) 1907 (d) 1901
38. 38.What is the units digit of 7^46? (a) 7 (b) 9 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 4? (a) 424 (b) 423 (c) 429 (d) 426
40. 40.Which of these numbers is divisible by 6? (a) 773 (b) 783 (c) 776 (d) 774

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