Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 575 · 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.Which of these numbers is divisible by 9? (a) 3313 (b) 3308 (c) 3303 (d) 3300
2. 2.Which of these numbers is divisible by 3? (a) 605 (b) 608 (c) 611 (d) 606
3. 3.What is the units digit of 4^51? (a) 5 (b) 6 (c) 4 (d) 3
4. 4.What is the units digit of 3^26? (a) 3 (b) 1 (c) 9 (d) 7
5. 5.Which of these numbers is divisible by 9? (a) 7857 (b) 7867 (c) 7863 (d) 7851
6. 6.What is the remainder when 179 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 3
7. 7.What is the remainder when 216 is divided by 11? (a) 7 (b) 8 (c) 4 (d) 6
8. 8.Which of these numbers is divisible by 6? (a) 1602 (b) 1597 (c) 1611 (d) 1604
9. 9.What is the smallest number you must add to 229 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 4
10. 10.Which of these numbers is divisible by 4? (a) 2047 (b) 2049 (c) 2048 (d) 2054
11. 11.What is the smallest number you must add to 463 to make it divisible by 11? (a) 9 (b) 8 (c) 1 (d) 10
12. 12.What is the remainder when 602 is divided by 11? (a) 7 (b) 3 (c) 9 (d) 8
13. 13.What is the smallest number you must add to 51 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
14. 14.What is the remainder when 346 is divided by 11? (a) 6 (b) 7 (c) 4 (d) 5
15. 15.What is the units digit of 4^55? (a) 6 (b) 3 (c) 5 (d) 4
16. 16.Which of these numbers is divisible by 8? (a) 4721 (b) 4735 (c) 4719 (d) 4720
17. 17.Which of these numbers is divisible by 11? (a) 660 (b) 657 (c) 678 (d) 662
18. 18.What is the units digit of 9^29? (a) 1 (b) 8 (c) 9 (d) 7
19. 19.What is the units digit of 8^30? (a) 2 (b) 8 (c) 6 (d) 4
20. 20.What is the units digit of 4^18? (a) 4 (b) 6 (c) 7 (d) 5
21. 21.What is the units digit of 2^25? (a) 6 (b) 4 (c) 2 (d) 8
22. 22.What is the smallest number you must add to 81 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
23. 23.Which of these numbers is divisible by 3? (a) 1721 (b) 1722 (c) 1723 (d) 1726
24. 24.What is the remainder when 523 is divided by 9? (a) 8 (b) 2 (c) 1 (d) 0
25. 25.What is the units digit of 2^13? (a) 6 (b) 2 (c) 4 (d) 8
26. 26.What is the smallest number you must add to 125 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 4
27. 27.What is the smallest number you must add to 801 to make it divisible by 11? (a) 1 (b) 2 (c) 9 (d) 3
28. 28.What is the remainder when 199 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
29. 29.What is the units digit of 3^11? (a) 3 (b) 9 (c) 7 (d) 1
30. 30.What is the smallest number you must add to 252 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 2
31. 31.Which of these numbers is divisible by 6? (a) 4577 (b) 4566 (c) 4563 (d) 4568
32. 32.What is the remainder when 73 is divided by 6? (a) 1 (b) 0 (c) 5 (d) 2
33. 33.What is the remainder when 353 is divided by 11? (a) 0 (b) 2 (c) 10 (d) 1
34. 34.What is the units digit of 2^14? (a) 8 (b) 2 (c) 6 (d) 4
35. 35.Which of these numbers is divisible by 8? (a) 2695 (b) 2688 (c) 2698 (d) 2687
36. 36.What is the units digit of 9^22? (a) 9 (b) 2 (c) 0 (d) 1
37. 37.Which of these numbers is divisible by 6? (a) 2413 (b) 2428 (c) 2418 (d) 2420
38. 38.What is the units digit of 7^29? (a) 7 (b) 1 (c) 3 (d) 9
39. 39.Which of these numbers is divisible by 6? (a) 5094 (b) 5095 (c) 5092 (d) 5105
40. 40.What is the remainder when 433 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1

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