Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 215 · 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 697 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 3
2. 2.What is the smallest number you must add to 266 to make it divisible by 4? (a) 2 (b) 3 (c) 1
3. 3.What is the smallest number you must add to 588 to make it divisible by 11? (a) 5 (b) 6 (c) 7 (d) 8
4. 4.Which of these numbers is divisible by 11? (a) 9451 (b) 9477 (c) 9461 (d) 9460
5. 5.Which of these numbers is divisible by 9? (a) 4207 (b) 4212 (c) 4222 (d) 4214
6. 6.What is the units digit of 9^11? (a) 1 (b) 9 (c) 7 (d) 8
7. 7.Which of these numbers is divisible by 8? (a) 3225 (b) 3209 (c) 3216 (d) 3222
8. 8.Which of these numbers is divisible by 11? (a) 4014 (b) 4029 (c) 4015 (d) 4016
9. 9.What is the units digit of 9^29? (a) 8 (b) 7 (c) 9 (d) 1
10. 10.What is the units digit of 8^50? (a) 8 (b) 6 (c) 2 (d) 4
11. 11.What is the units digit of 4^60? (a) 4 (b) 7 (c) 6 (d) 5
12. 12.What is the remainder when 65 is divided by 3? (a) 0 (b) 2 (c) 1
13. 13.Which of these numbers is divisible by 11? (a) 8834 (b) 8845 (c) 8826 (d) 8833
14. 14.Which of these numbers is divisible by 3? (a) 2075 (b) 2076 (c) 2081 (d) 2077
15. 15.What is the units digit of 3^50? (a) 7 (b) 3 (c) 9 (d) 1
16. 16.What is the remainder when 213 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
17. 17.What is the remainder when 439 is divided by 11? (a) 9 (b) 8 (c) 1 (d) 10
18. 18.What is the smallest number you must add to 419 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 1
19. 19.What is the remainder when 383 is divided by 10? (a) 4 (b) 2 (c) 7 (d) 3
20. 20.What is the units digit of 7^53? (a) 9 (b) 1 (c) 7 (d) 3
21. 21.What is the smallest number you must add to 302 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
22. 22.Which of these numbers is divisible by 3? (a) 2432 (b) 2434 (c) 2438 (d) 2433
23. 23.Which of these numbers is divisible by 4? (a) 300 (b) 298 (c) 306 (d) 303
24. 24.What is the remainder when 41 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
25. 25.What is the remainder when 170 is divided by 6? (a) 2 (b) 3 (c) 1 (d) 4
26. 26.What is the remainder when 568 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 671 (b) 677 (c) 672 (d) 673
28. 28.Which of these numbers is divisible by 8? (a) 4240 (b) 4238 (c) 4251 (d) 4242
29. 29.What is the units digit of 7^28? (a) 1 (b) 9 (c) 7 (d) 3
30. 30.Which of these numbers is divisible by 3? (a) 1409 (b) 1412 (c) 1407 (d) 1406
31. 31.What is the smallest number you must add to 653 to make it divisible by 11? (a) 8 (b) 7 (c) 4 (d) 6
32. 32.What is the remainder when 266 is divided by 11? (a) 3 (b) 9 (c) 2 (d) 1
33. 33.What is the units digit of 2^46? (a) 8 (b) 4 (c) 2 (d) 6
34. 34.What is the smallest number you must add to 178 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 3
35. 35.Which of these numbers is divisible by 8? (a) 921 (b) 909 (c) 912 (d) 913
36. 36.What is the remainder when 206 is divided by 11? (a) 7 (b) 9 (c) 8 (d) 3
37. 37.What is the units digit of 8^34? (a) 4 (b) 6 (c) 8 (d) 2
38. 38.Which of these numbers is divisible by 8? (a) 6528 (b) 6522 (c) 6540 (d) 6533
39. 39.Which of these numbers is divisible by 3? (a) 2624 (b) 2625 (c) 2627 (d) 2630
40. 40.What is the units digit of 7^11? (a) 9 (b) 7 (c) 1 (d) 3

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