Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 675 · 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 3^57? (a) 1 (b) 7 (c) 9 (d) 3
2. 2.What is the remainder when 443 is divided by 11? (a) 2 (b) 3 (c) 8 (d) 4
3. 3.What is the remainder when 97 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
4. 4.Which of these numbers is divisible by 9? (a) 4939 (b) 4944 (c) 4955 (d) 4941
5. 5.What is the remainder when 480 is divided by 11? (a) 4 (b) 6 (c) 8 (d) 7
6. 6.Which of these numbers is divisible by 4? (a) 3341 (b) 3345 (c) 3340 (d) 3338
7. 7.What is the units digit of 8^46? (a) 6 (b) 8 (c) 2 (d) 4
8. 8.Which of these numbers is divisible by 9? (a) 2579 (b) 2584 (c) 2583 (d) 2593
9. 9.What is the smallest number you must add to 216 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
10. 10.What is the units digit of 4^14? (a) 7 (b) 6 (c) 5 (d) 4
11. 11.Which of these numbers is divisible by 9? (a) 4709 (b) 4695 (c) 4698 (d) 4700
12. 12.What is the remainder when 97 is divided by 3? (a) 0 (b) 2 (c) 1
13. 13.What is the smallest number you must add to 143 to make it divisible by 12? (a) 2 (b) 11 (c) 3 (d) 1
14. 14.What is the remainder when 318 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
15. 15.What is the smallest number you must add to 438 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1
16. 16.What is the units digit of 8^17? (a) 8 (b) 4 (c) 6 (d) 2
17. 17.What is the units digit of 7^16? (a) 7 (b) 9 (c) 1 (d) 3
18. 18.What is the smallest number you must add to 244 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 7
19. 19.What is the units digit of 3^35? (a) 1 (b) 9 (c) 7 (d) 3
20. 20.What is the remainder when 432 is divided by 11? (a) 8 (b) 4 (c) 2 (d) 3
21. 21.Which of these numbers is divisible by 3? (a) 697 (b) 692 (c) 693 (d) 694
22. 22.Which of these numbers is divisible by 6? (a) 3733 (b) 3726 (c) 3725 (d) 3731
23. 23.Which of these numbers is divisible by 11? (a) 8886 (b) 8898 (c) 8877 (d) 8875
24. 24.What is the remainder when 466 is divided by 8? (a) 2 (b) 6 (c) 3 (d) 1
25. 25.Which of these numbers is divisible by 9? (a) 1286 (b) 1277 (c) 1268 (d) 1269
26. 26.What is the units digit of 8^18? (a) 2 (b) 6 (c) 8 (d) 4
27. 27.What is the smallest number you must add to 676 to make it divisible by 9? (a) 1 (b) 6 (c) 8 (d) 7
28. 28.Which of these numbers is divisible by 9? (a) 1852 (b) 1854 (c) 1871 (d) 1856
29. 29.What is the smallest number you must add to 531 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
30. 30.What is the smallest number you must add to 258 to make it divisible by 10? (a) 2 (b) 1 (c) 3 (d) 8
31. 31.What is the remainder when 122 is divided by 11? (a) 2 (b) 10 (c) 1 (d) 0
32. 32.What is the smallest number you must add to 101 to make it divisible by 4? (a) 2 (b) 1 (c) 3
33. 33.What is the remainder when 283 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
34. 34.What is the smallest number you must add to 626 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
35. 35.What is the units digit of 9^32? (a) 1 (b) 0 (c) 9 (d) 2
36. 36.Which of these numbers is divisible by 3? (a) 128 (b) 122 (c) 123 (d) 125
37. 37.What is the units digit of 3^30? (a) 1 (b) 7 (c) 3 (d) 9
38. 38.What is the units digit of 3^25? (a) 3 (b) 7 (c) 1 (d) 9
39. 39.What is the remainder when 314 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
40. 40.Which of these numbers is divisible by 9? (a) 2022 (b) 2041 (c) 2032 (d) 2025

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