Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 310 · 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 222 is divided by 8? (a) 5 (b) 2 (c) 6 (d) 7
2. 2.What is the smallest number you must add to 244 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
3. 3.What is the remainder when 187 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
4. 4.Which of these numbers is divisible by 3? (a) 121 (b) 124 (c) 123 (d) 128
5. 5.What is the smallest number you must add to 431 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 2
6. 6.What is the remainder when 448 is divided by 9? (a) 2 (b) 6 (c) 7 (d) 8
7. 7.What is the units digit of 9^41? (a) 8 (b) 1 (c) 9 (d) 7
8. 8.Which of these numbers is divisible by 8? (a) 1473 (b) 1469 (c) 1483 (d) 1472
9. 9.Which of these numbers is divisible by 11? (a) 1040 (b) 1045 (c) 1064 (d) 1051
10. 10.What is the smallest number you must add to 567 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 4
11. 11.What is the smallest number you must add to 417 to make it divisible by 12? (a) 2 (b) 9 (c) 3 (d) 4
12. 12.What is the remainder when 446 is divided by 12? (a) 3 (b) 1 (c) 2 (d) 10
13. 13.What is the remainder when 396 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
14. 14.What is the units digit of 9^59? (a) 8 (b) 1 (c) 7 (d) 9
15. 15.What is the smallest number you must add to 155 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
16. 16.What is the units digit of 2^26? (a) 6 (b) 8 (c) 2 (d) 4
17. 17.What is the remainder when 742 is divided by 10? (a) 3 (b) 2 (c) 8 (d) 1
18. 18.Which of these numbers is divisible by 9? (a) 408 (b) 405 (c) 403 (d) 415
19. 19.Which of these numbers is divisible by 3? (a) 2488 (b) 2485 (c) 2492 (d) 2487
20. 20.What is the remainder when 81 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
21. 21.Which of these numbers is divisible by 4? (a) 1556 (b) 1555 (c) 1557 (d) 1561
22. 22.Which of these numbers is divisible by 8? (a) 2496 (b) 2490 (c) 2506 (d) 2503
23. 23.What is the smallest number you must add to 116 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the units digit of 2^18? (a) 2 (b) 6 (c) 4 (d) 8
25. 25.What is the units digit of 9^26? (a) 2 (b) 0 (c) 1 (d) 9
26. 26.What is the units digit of 7^26? (a) 3 (b) 9 (c) 7 (d) 1
27. 27.What is the remainder when 584 is divided by 11? (a) 10 (b) 0 (c) 1 (d) 2
28. 28.What is the remainder when 247 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
29. 29.Which of these numbers is divisible by 4? (a) 243 (b) 244 (c) 245 (d) 249
30. 30.What is the remainder when 228 is divided by 9? (a) 2 (b) 4 (c) 3 (d) 6
31. 31.What is the units digit of 2^46? (a) 6 (b) 4 (c) 8 (d) 2
32. 32.What is the remainder when 101 is divided by 3? (a) 0 (b) 1 (c) 2
33. 33.What is the smallest number you must add to 241 to make it divisible by 4? (a) 3 (b) 1 (c) 2
34. 34.What is the smallest number you must add to 369 to make it divisible by 10? (a) 2 (b) 9 (c) 3 (d) 1
35. 35.What is the smallest number you must add to 337 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
36. 36.What is the remainder when 583 is divided by 10? (a) 4 (b) 7 (c) 3 (d) 2
37. 37.What is the remainder when 424 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
38. 38.What is the units digit of 2^47? (a) 6 (b) 4 (c) 8 (d) 2
39. 39.Which of these numbers is divisible by 4? (a) 3423 (b) 3417 (c) 3427 (d) 3420
40. 40.Which of these numbers is divisible by 3? (a) 482 (b) 477 (c) 478 (d) 476

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