Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 742 · 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 8? (a) 2636 (b) 2625 (c) 2624 (d) 2622
2. 2.What is the remainder when 309 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 6
3. 3.What is the remainder when 586 is divided by 12? (a) 2 (b) 11 (c) 9 (d) 10
4. 4.What is the units digit of 7^49? (a) 9 (b) 1 (c) 7 (d) 3
5. 5.What is the smallest number you must add to 615 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
6. 6.Which of these numbers is divisible by 8? (a) 5828 (b) 5813 (c) 5822 (d) 5816
7. 7.What is the smallest number you must add to 113 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 5
8. 8.What is the remainder when 104 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
9. 9.What is the remainder when 874 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 6
10. 10.What is the units digit of 2^19? (a) 6 (b) 8 (c) 2 (d) 4
11. 11.What is the units digit of 4^39? (a) 4 (b) 6 (c) 5 (d) 3
12. 12.What is the smallest number you must add to 168 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
13. 13.Which of these numbers is divisible by 3? (a) 550 (b) 546 (c) 548 (d) 544
14. 14.What is the smallest number you must add to 156 to make it divisible by 7? (a) 5 (b) 6 (c) 2 (d) 4
15. 15.What is the units digit of 8^54? (a) 8 (b) 2 (c) 6 (d) 4
16. 16.What is the smallest number you must add to 200 to make it divisible by 11? (a) 10 (b) 9 (c) 2 (d) 8
17. 17.What is the smallest number you must add to 175 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
18. 18.Which of these numbers is divisible by 6? (a) 673 (b) 670 (c) 666 (d) 665
19. 19.Which of these numbers is divisible by 3? (a) 1843 (b) 1839 (c) 1841 (d) 1838
20. 20.What is the remainder when 595 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
21. 21.Which of these numbers is divisible by 9? (a) 7489 (b) 7486 (c) 7479 (d) 7477
22. 22.What is the remainder when 588 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
23. 23.What is the remainder when 479 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 8
24. 24.What is the remainder when 177 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
25. 25.What is the smallest number you must add to 77 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 2
26. 26.What is the remainder when 280 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
27. 27.What is the units digit of 7^30? (a) 1 (b) 7 (c) 3 (d) 9
28. 28.What is the units digit of 3^25? (a) 3 (b) 7 (c) 9 (d) 1
29. 29.What is the remainder when 809 is divided by 10? (a) 9 (b) 7 (c) 1 (d) 8
30. 30.What is the smallest number you must add to 68 to make it divisible by 3? (a) 2 (b) 1
31. 31.What is the units digit of 9^50? (a) 9 (b) 2 (c) 0 (d) 1
32. 32.What is the remainder when 142 is divided by 3? (a) 1 (b) 2 (c) 0
33. 33.What is the units digit of 9^36? (a) 0 (b) 1 (c) 2 (d) 9
34. 34.Which of these numbers is divisible by 6? (a) 5044 (b) 5034 (c) 5036 (d) 5033
35. 35.What is the remainder when 724 is divided by 12? (a) 5 (b) 3 (c) 8 (d) 4
36. 36.What is the remainder when 271 is divided by 9? (a) 0 (b) 8 (c) 2 (d) 1
37. 37.What is the remainder when 542 is divided by 8? (a) 6 (b) 7 (c) 2 (d) 5
38. 38.What is the units digit of 9^59? (a) 9 (b) 8 (c) 1 (d) 7
39. 39.Which of these numbers is divisible by 6? (a) 4574 (b) 4569 (c) 4572 (d) 4580
40. 40.What is the remainder when 71 is divided by 3? (a) 1 (b) 2 (c) 0

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