Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 827 · 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 140 is divided by 11? (a) 9 (b) 7 (c) 3 (d) 8
2. 2.Which of these numbers is divisible by 6? (a) 4780 (b) 4783 (c) 4773 (d) 4776
3. 3.What is the smallest number you must add to 505 to make it divisible by 7? (a) 5 (b) 6 (c) 1 (d) 4
4. 4.What is the remainder when 354 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
5. 5.Which of these numbers is divisible by 4? (a) 612 (b) 613 (c) 618 (d) 611
6. 6.What is the remainder when 286 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
7. 7.Which of these numbers is divisible by 4? (a) 1705 (b) 1700 (c) 1697 (d) 1701
8. 8.What is the remainder when 676 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
9. 9.What is the units digit of 8^10? (a) 4 (b) 2 (c) 8 (d) 6
10. 10.Which of these numbers is divisible by 3? (a) 876 (b) 878 (c) 881 (d) 875
11. 11.Which of these numbers is divisible by 9? (a) 7627 (b) 7616 (c) 7640 (d) 7623
12. 12.Which of these numbers is divisible by 6? (a) 1744 (b) 1740 (c) 1737 (d) 1748
13. 13.What is the units digit of 9^45? (a) 9 (b) 1 (c) 7 (d) 8
14. 14.What is the smallest number you must add to 69 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
15. 15.What is the remainder when 364 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
16. 16.Which of these numbers is divisible by 11? (a) 2236 (b) 2230 (c) 2245 (d) 2233
17. 17.Which of these numbers is divisible by 6? (a) 4903 (b) 4913 (c) 4897 (d) 4902
18. 18.What is the smallest number you must add to 734 to make it divisible by 12? (a) 11 (b) 10 (c) 2 (d) 9
19. 19.Which of these numbers is divisible by 6? (a) 1027 (b) 1026 (c) 1034 (d) 1023
20. 20.What is the units digit of 7^59? (a) 9 (b) 1 (c) 3 (d) 7
21. 21.What is the smallest number you must add to 238 to make it divisible by 3? (a) 2 (b) 1
22. 22.Which of these numbers is divisible by 9? (a) 6714 (b) 6706 (c) 6728 (d) 6720
23. 23.Which of these numbers is divisible by 3? (a) 385 (b) 384 (c) 389 (d) 382
24. 24.What is the smallest number you must add to 244 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 1
25. 25.Which of these numbers is divisible by 4? (a) 1536 (b) 1534 (c) 1539 (d) 1543
26. 26.What is the smallest number you must add to 209 to make it divisible by 3? (a) 1 (b) 2
27. 27.What is the units digit of 3^42? (a) 7 (b) 1 (c) 3 (d) 9
28. 28.Which of these numbers is divisible by 8? (a) 1025 (b) 1014 (c) 1016 (d) 1019
29. 29.What is the smallest number you must add to 352 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 2
30. 30.Which of these numbers is divisible by 9? (a) 1566 (b) 1583 (c) 1570 (d) 1563
31. 31.What is the remainder when 223 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
32. 32.What is the smallest number you must add to 357 to make it divisible by 12? (a) 4 (b) 3 (c) 2 (d) 9
33. 33.What is the units digit of 3^51? (a) 3 (b) 1 (c) 7 (d) 9
34. 34.What is the smallest number you must add to 736 to make it divisible by 11? (a) 2 (b) 3 (c) 10 (d) 1
35. 35.What is the smallest number you must add to 318 to make it divisible by 4? (a) 1 (b) 2 (c) 3
36. 36.What is the remainder when 295 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
37. 37.What is the smallest number you must add to 188 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
38. 38.What is the units digit of 2^11? (a) 2 (b) 8 (c) 6 (d) 4
39. 39.What is the remainder when 314 is divided by 10? (a) 5 (b) 3 (c) 4 (d) 6
40. 40.What is the remainder when 139 is divided by 8? (a) 3 (b) 2 (c) 4 (d) 5

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