Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 993 · 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 429 is divided by 10? (a) 1 (b) 8 (c) 7 (d) 9
2. 2.Which of these numbers is divisible by 8? (a) 5788 (b) 5771 (c) 5781 (d) 5776
3. 3.What is the remainder when 797 is divided by 12? (a) 5 (b) 7 (c) 6 (d) 4
4. 4.Which of these numbers is divisible by 4? (a) 3388 (b) 3389 (c) 3395 (d) 3386
5. 5.What is the remainder when 520 is divided by 12? (a) 5 (b) 3 (c) 4 (d) 8
6. 6.What is the remainder when 653 is divided by 9? (a) 7 (b) 4 (c) 6 (d) 5
7. 7.What is the units digit of 8^59? (a) 8 (b) 4 (c) 2 (d) 6
8. 8.Which of these numbers is divisible by 4? (a) 647 (b) 639 (c) 640 (d) 641
9. 9.What is the units digit of 2^32? (a) 2 (b) 6 (c) 8 (d) 4
10. 10.What is the remainder when 242 is divided by 3? (a) 1 (b) 2 (c) 0
11. 11.What is the remainder when 587 is divided by 9? (a) 1 (b) 7 (c) 2 (d) 3
12. 12.What is the smallest number you must add to 333 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
13. 13.What is the smallest number you must add to 426 to make it divisible by 9? (a) 3 (b) 5 (c) 7 (d) 6
14. 14.Which of these numbers is divisible by 3? (a) 755 (b) 751 (c) 748 (d) 750
15. 15.What is the units digit of 3^12? (a) 9 (b) 3 (c) 1 (d) 7
16. 16.What is the units digit of 2^11? (a) 4 (b) 6 (c) 8 (d) 2
17. 17.What is the remainder when 425 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
18. 18.Which of these numbers is divisible by 4? (a) 792 (b) 793 (c) 791 (d) 798
19. 19.What is the units digit of 9^55? (a) 1 (b) 9 (c) 7 (d) 8
20. 20.Which of these numbers is divisible by 6? (a) 3618 (b) 3615 (c) 3628 (d) 3620
21. 21.What is the remainder when 157 is divided by 12? (a) 1 (b) 2 (c) 0 (d) 11
22. 22.Which of these numbers is divisible by 8? (a) 3791 (b) 3775 (c) 3776 (d) 3779
23. 23.What is the smallest number you must add to 190 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the remainder when 95 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
25. 25.What is the units digit of 8^29? (a) 8 (b) 6 (c) 4 (d) 2
26. 26.Which of these numbers is divisible by 8? (a) 1930 (b) 1921 (c) 1928 (d) 1942
27. 27.What is the smallest number you must add to 370 to make it divisible by 7? (a) 6 (b) 1 (c) 2 (d) 3
28. 28.What is the remainder when 420 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 3
29. 29.What is the remainder when 559 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 8
30. 30.Which of these numbers is divisible by 4? (a) 3456 (b) 3455 (c) 3462 (d) 3458
31. 31.Which of these numbers is divisible by 9? (a) 1334 (b) 1323 (c) 1330 (d) 1321
32. 32.What is the smallest number you must add to 158 to make it divisible by 4? (a) 2 (b) 1 (c) 3
33. 33.Which of these numbers is divisible by 3? (a) 316 (b) 315 (c) 320 (d) 314
34. 34.Which of these numbers is divisible by 4? (a) 2773 (b) 2769 (c) 2766 (d) 2768
35. 35.What is the smallest number you must add to 83 to make it divisible by 4? (a) 2 (b) 1 (c) 3
36. 36.What is the units digit of 2^56? (a) 2 (b) 4 (c) 6 (d) 8
37. 37.Which of these numbers is divisible by 6? (a) 4885 (b) 4893 (c) 4901 (d) 4890
38. 38.What is the remainder when 151 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
39. 39.What is the units digit of 2^37? (a) 2 (b) 6 (c) 8 (d) 4
40. 40.Which of these numbers is divisible by 4? (a) 3425 (b) 3433 (c) 3430 (d) 3428

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