Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 261 · 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 510 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 7
2. 2.Which of these numbers is divisible by 6? (a) 544 (b) 557 (c) 546 (d) 551
3. 3.What is the smallest number you must add to 289 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
4. 4.What is the units digit of 9^53? (a) 7 (b) 1 (c) 9 (d) 8
5. 5.Which of these numbers is divisible by 8? (a) 576 (b) 590 (c) 575 (d) 583
6. 6.What is the remainder when 428 is divided by 7? (a) 2 (b) 6 (c) 1 (d) 0
7. 7.What is the units digit of 2^25? (a) 4 (b) 2 (c) 6 (d) 8
8. 8.What is the smallest number you must add to 277 to make it divisible by 9? (a) 7 (b) 2 (c) 1 (d) 3
9. 9.Which of these numbers is divisible by 6? (a) 2913 (b) 2902 (c) 2904 (d) 2906
10. 10.Which of these numbers is divisible by 3? (a) 1261 (b) 1263 (c) 1265 (d) 1267
11. 11.What is the smallest number you must add to 68 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
12. 12.Which of these numbers is divisible by 11? (a) 3569 (b) 3558 (c) 3578 (d) 3564
13. 13.What is the remainder when 530 is divided by 11? (a) 9 (b) 2 (c) 1 (d) 3
14. 14.Which of these numbers is divisible by 8? (a) 3072 (b) 3075 (c) 3081 (d) 3071
15. 15.What is the remainder when 966 is divided by 12? (a) 5 (b) 8 (c) 7 (d) 6
16. 16.Which of these numbers is divisible by 6? (a) 1531 (b) 1521 (c) 1528 (d) 1524
17. 17.What is the units digit of 7^37? (a) 7 (b) 9 (c) 3 (d) 1
18. 18.What is the remainder when 845 is divided by 11? (a) 8 (b) 2 (c) 9 (d) 10
19. 19.What is the remainder when 576 is divided by 11? (a) 7 (b) 5 (c) 4 (d) 3
20. 20.Which of these numbers is divisible by 3? (a) 2354 (b) 2352 (c) 2351 (d) 2356
21. 21.What is the smallest number you must add to 173 to make it divisible by 3? (a) 1 (b) 2
22. 22.Which of these numbers is divisible by 3? (a) 574 (b) 570 (c) 569 (d) 572
23. 23.Which of these numbers is divisible by 8? (a) 5826 (b) 5822 (c) 5816 (d) 5812
24. 24.What is the remainder when 515 is divided by 11? (a) 10 (b) 8 (c) 9 (d) 2
25. 25.What is the remainder when 203 is divided by 3? (a) 0 (b) 1 (c) 2
26. 26.Which of these numbers is divisible by 6? (a) 342 (b) 339 (c) 344 (d) 353
27. 27.What is the remainder when 338 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
28. 28.What is the remainder when 158 is divided by 10? (a) 7 (b) 9 (c) 2 (d) 8
29. 29.Which of these numbers is divisible by 11? (a) 7986 (b) 7982 (c) 7988 (d) 8006
30. 30.Which of these numbers is divisible by 9? (a) 961 (b) 978 (c) 966 (d) 963
31. 31.Which of these numbers is divisible by 4? (a) 859 (b) 862 (c) 856 (d) 854
32. 32.Which of these numbers is divisible by 3? (a) 729 (b) 733 (c) 730 (d) 727
33. 33.Which of these numbers is divisible by 4? (a) 888 (b) 895 (c) 891 (d) 885
34. 34.What is the units digit of 4^51? (a) 6 (b) 5 (c) 4 (d) 3
35. 35.What is the smallest number you must add to 372 to make it divisible by 10? (a) 7 (b) 8 (c) 2 (d) 9
36. 36.Which of these numbers is divisible by 11? (a) 759 (b) 780 (c) 766 (d) 754
37. 37.What is the units digit of 4^15? (a) 4 (b) 5 (c) 6 (d) 3
38. 38.What is the smallest number you must add to 303 to make it divisible by 12? (a) 10 (b) 8 (c) 9 (d) 3
39. 39.What is the units digit of 3^25? (a) 1 (b) 7 (c) 3 (d) 9
40. 40.What is the remainder when 248 is divided by 10? (a) 8 (b) 7 (c) 9 (d) 2

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