Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 397 · 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 3^51? (a) 7 (b) 1 (c) 3 (d) 9
2. 2.Which of these numbers is divisible by 11? (a) 895 (b) 883 (c) 891 (d) 907
3. 3.What is the units digit of 2^26? (a) 2 (b) 8 (c) 4 (d) 6
4. 4.Which of these numbers is divisible by 9? (a) 6922 (b) 6921 (c) 6920 (d) 6933
5. 5.What is the remainder when 359 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
6. 6.What is the units digit of 4^25? (a) 6 (b) 3 (c) 4 (d) 5
7. 7.What is the smallest number you must add to 474 to make it divisible by 8? (a) 2 (b) 6 (c) 7 (d) 5
8. 8.Which of these numbers is divisible by 9? (a) 3549 (b) 3537 (c) 3542 (d) 3535
9. 9.Which of these numbers is divisible by 6? (a) 2521 (b) 2516 (c) 2514 (d) 2512
10. 10.Which of these numbers is divisible by 11? (a) 7326 (b) 7322 (c) 7328 (d) 7340
11. 11.What is the units digit of 9^12? (a) 0 (b) 1 (c) 2 (d) 9
12. 12.What is the smallest number you must add to 107 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
13. 13.Which of these numbers is divisible by 4? (a) 473 (b) 469 (c) 472 (d) 478
14. 14.What is the remainder when 108 is divided by 7? (a) 2 (b) 3 (c) 4 (d) 5
15. 15.Which of these numbers is divisible by 9? (a) 3738 (b) 3726 (c) 3725 (d) 3727
16. 16.What is the remainder when 305 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
17. 17.What is the remainder when 482 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
18. 18.What is the remainder when 340 is divided by 9? (a) 7 (b) 8 (c) 2 (d) 6
19. 19.Which of these numbers is divisible by 3? (a) 571 (b) 574 (c) 570 (d) 569
20. 20.What is the smallest number you must add to 211 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
21. 21.What is the remainder when 206 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
22. 22.What is the units digit of 2^34? (a) 2 (b) 4 (c) 6 (d) 8
23. 23.What is the remainder when 93 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
24. 24.Which of these numbers is divisible by 6? (a) 2461 (b) 2475 (c) 2466 (d) 2467
25. 25.What is the smallest number you must add to 172 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the remainder when 703 is divided by 11? (a) 8 (b) 9 (c) 1 (d) 10
27. 27.What is the units digit of 2^59? (a) 8 (b) 6 (c) 2 (d) 4
28. 28.What is the remainder when 636 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
29. 29.Which of these numbers is divisible by 11? (a) 2411 (b) 2420 (c) 2426 (d) 2441
30. 30.What is the remainder when 301 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
31. 31.What is the remainder when 148 is divided by 3? (a) 2 (b) 1 (c) 0
32. 32.What is the remainder when 299 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
33. 33.Which of these numbers is divisible by 8? (a) 373 (b) 354 (c) 360 (d) 366
34. 34.What is the remainder when 318 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
35. 35.What is the remainder when 373 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
36. 36.Which of these numbers is divisible by 11? (a) 9873 (b) 9878 (c) 9892 (d) 9885
37. 37.Which of these numbers is divisible by 4? (a) 3591 (b) 3593 (c) 3598 (d) 3592
38. 38.Which of these numbers is divisible by 6? (a) 2580 (b) 2578 (c) 2581 (d) 2589
39. 39.What is the remainder when 218 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
40. 40.Which of these numbers is divisible by 6? (a) 4576 (b) 4579 (c) 4578 (d) 4586

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