Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 559 · 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) 3446 (b) 3447 (c) 3458 (d) 3449
2. 2.Which of these numbers is divisible by 9? (a) 1893 (b) 1887 (c) 1890 (d) 1904
3. 3.What is the remainder when 166 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
4. 4.What is the remainder when 139 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
5. 5.What is the smallest number you must add to 545 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
6. 6.What is the units digit of 7^58? (a) 3 (b) 7 (c) 9 (d) 1
7. 7.Which of these numbers is divisible by 9? (a) 6882 (b) 6867 (c) 6864 (d) 6868
8. 8.What is the smallest number you must add to 218 to make it divisible by 4? (a) 1 (b) 2 (c) 3
9. 9.Which of these numbers is divisible by 4? (a) 3533 (b) 3537 (c) 3541 (d) 3536
10. 10.Which of these numbers is divisible by 3? (a) 523 (b) 527 (c) 530 (d) 525
11. 11.Which of these numbers is divisible by 4? (a) 1026 (b) 1030 (c) 1028 (d) 1033
12. 12.What is the smallest number you must add to 71 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the smallest number you must add to 426 to make it divisible by 11? (a) 2 (b) 4 (c) 8 (d) 3
14. 14.What is the smallest number you must add to 99 to make it divisible by 7? (a) 1 (b) 6 (c) 5 (d) 4
15. 15.What is the units digit of 4^33? (a) 4 (b) 3 (c) 5 (d) 6
16. 16.Which of these numbers is divisible by 6? (a) 2859 (b) 2845 (c) 2850 (d) 2851
17. 17.What is the units digit of 8^60? (a) 4 (b) 8 (c) 2 (d) 6
18. 18.Which of these numbers is divisible by 4? (a) 2726 (b) 2724 (c) 2729 (d) 2721
19. 19.What is the remainder when 535 is divided by 9? (a) 3 (b) 5 (c) 4 (d) 6
20. 20.What is the units digit of 9^43? (a) 1 (b) 9 (c) 7 (d) 8
21. 21.What is the remainder when 135 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
22. 22.Which of these numbers is divisible by 8? (a) 3828 (b) 3838 (c) 3844 (d) 3832
23. 23.Which of these numbers is divisible by 6? (a) 634 (b) 623 (c) 624 (d) 629
24. 24.What is the units digit of 3^55? (a) 9 (b) 3 (c) 1 (d) 7
25. 25.What is the remainder when 211 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
26. 26.What is the remainder when 173 is divided by 9? (a) 7 (b) 3 (c) 2 (d) 1
27. 27.What is the remainder when 566 is divided by 12? (a) 3 (b) 10 (c) 2 (d) 1
28. 28.Which of these numbers is divisible by 11? (a) 9500 (b) 9519 (c) 9510 (d) 9504
29. 29.What is the units digit of 9^15? (a) 1 (b) 8 (c) 9 (d) 7
30. 30.Which of these numbers is divisible by 8? (a) 6521 (b) 6512 (c) 6516 (d) 6511
31. 31.What is the units digit of 3^56? (a) 1 (b) 7 (c) 3 (d) 9
32. 32.Which of these numbers is divisible by 4? (a) 1695 (b) 1691 (c) 1688 (d) 1686
33. 33.What is the units digit of 3^58? (a) 1 (b) 9 (c) 3 (d) 7
34. 34.What is the units digit of 2^43? (a) 2 (b) 4 (c) 8 (d) 6
35. 35.What is the smallest number you must add to 263 to make it divisible by 8? (a) 3 (b) 1 (c) 2 (d) 7
36. 36.What is the smallest number you must add to 741 to make it divisible by 10? (a) 8 (b) 1 (c) 9 (d) 7
37. 37.What is the units digit of 4^18? (a) 4 (b) 6 (c) 5 (d) 7
38. 38.What is the remainder when 182 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.What is the units digit of 2^26? (a) 2 (b) 4 (c) 6 (d) 8
40. 40.Which of these numbers is divisible by 8? (a) 603 (b) 594 (c) 611 (d) 600

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