Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 956 · 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 remainder when 244 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
2. 2.What is the remainder when 152 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
3. 3.What is the units digit of 2^13? (a) 8 (b) 2 (c) 4 (d) 6
4. 4.Which of these numbers is divisible by 11? (a) 6166 (b) 6160 (c) 6155 (d) 6177
5. 5.What is the remainder when 664 is divided by 12? (a) 4 (b) 8 (c) 5 (d) 3
6. 6.What is the remainder when 119 is divided by 3? (a) 0 (b) 2 (c) 1
7. 7.What is the smallest number you must add to 179 to make it divisible by 4? (a) 2 (b) 1 (c) 3
8. 8.Which of these numbers is divisible by 11? (a) 684 (b) 682 (c) 695 (d) 673
9. 9.What is the units digit of 4^18? (a) 7 (b) 5 (c) 4 (d) 6
10. 10.What is the smallest number you must add to 97 to make it divisible by 4? (a) 3 (b) 1 (c) 2
11. 11.What is the remainder when 849 is divided by 11? (a) 3 (b) 1 (c) 2 (d) 9
12. 12.What is the units digit of 8^34? (a) 6 (b) 2 (c) 4 (d) 8
13. 13.What is the units digit of 3^18? (a) 3 (b) 7 (c) 9 (d) 1
14. 14.Which of these numbers is divisible by 4? (a) 804 (b) 810 (c) 802 (d) 806
15. 15.Which of these numbers is divisible by 8? (a) 1544 (b) 1541 (c) 1551 (d) 1557
16. 16.Which of these numbers is divisible by 4? (a) 1279 (b) 1280 (c) 1283 (d) 1285
17. 17.What is the remainder when 823 is divided by 12? (a) 8 (b) 5 (c) 7 (d) 6
18. 18.Which of these numbers is divisible by 4? (a) 642 (b) 637 (c) 640 (d) 645
19. 19.Which of these numbers is divisible by 8? (a) 3937 (b) 3928 (c) 3921 (d) 3931
20. 20.What is the smallest number you must add to 342 to make it divisible by 8? (a) 6 (b) 2 (c) 1 (d) 3
21. 21.Which of these numbers is divisible by 9? (a) 2752 (b) 2744 (c) 2731 (d) 2736
22. 22.What is the units digit of 2^43? (a) 6 (b) 8 (c) 2 (d) 4
23. 23.What is the units digit of 7^24? (a) 1 (b) 3 (c) 7 (d) 9
24. 24.What is the remainder when 112 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 4
25. 25.What is the remainder when 256 is divided by 9? (a) 3 (b) 6 (c) 4 (d) 5
26. 26.What is the remainder when 155 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
27. 27.Which of these numbers is divisible by 4? (a) 3511 (b) 3504 (c) 3501 (d) 3506
28. 28.What is the remainder when 251 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
29. 29.What is the units digit of 9^15? (a) 8 (b) 1 (c) 9 (d) 7
30. 30.What is the smallest number you must add to 155 to make it divisible by 4? (a) 2 (b) 1 (c) 3
31. 31.What is the smallest number you must add to 230 to make it divisible by 11? (a) 1 (b) 10 (c) 2 (d) 3
32. 32.What is the units digit of 4^60? (a) 6 (b) 5 (c) 7 (d) 4
33. 33.What is the units digit of 9^43? (a) 1 (b) 8 (c) 9 (d) 7
34. 34.What is the units digit of 3^46? (a) 7 (b) 3 (c) 9 (d) 1
35. 35.What is the remainder when 130 is divided by 3? (a) 1 (b) 0 (c) 2
36. 36.Which of these numbers is divisible by 4? (a) 2291 (b) 2286 (c) 2284 (d) 2282
37. 37.What is the units digit of 3^41? (a) 1 (b) 9 (c) 7 (d) 3
38. 38.Which of these numbers is divisible by 3? (a) 809 (b) 814 (c) 810 (d) 812
39. 39.Which of these numbers is divisible by 3? (a) 2195 (b) 2192 (c) 2197 (d) 2193
40. 40.What is the remainder when 507 is divided by 9? (a) 2 (b) 6 (c) 4 (d) 3

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