Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 756 · 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 436 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
2. 2.What is the remainder when 611 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 1
3. 3.What is the smallest number you must add to 95 to make it divisible by 9? (a) 6 (b) 5 (c) 3 (d) 4
4. 4.What is the units digit of 9^50? (a) 2 (b) 9 (c) 1 (d) 0
5. 5.What is the units digit of 3^47? (a) 1 (b) 3 (c) 7 (d) 9
6. 6.Which of these numbers is divisible by 3? (a) 133 (b) 140 (c) 136 (d) 135
7. 7.What is the smallest number you must add to 695 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 2
8. 8.What is the smallest number you must add to 556 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 3
9. 9.What is the remainder when 459 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
10. 10.What is the units digit of 2^36? (a) 2 (b) 8 (c) 4 (d) 6
11. 11.What is the units digit of 2^28? (a) 4 (b) 6 (c) 8 (d) 2
12. 12.What is the remainder when 173 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
13. 13.Which of these numbers is divisible by 8? (a) 3624 (b) 3639 (c) 3630 (d) 3617
14. 14.What is the units digit of 9^30? (a) 9 (b) 1 (c) 2 (d) 0
15. 15.Which of these numbers is divisible by 4? (a) 3106 (b) 3113 (c) 3109 (d) 3108
16. 16.What is the smallest number you must add to 376 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
17. 17.Which of these numbers is divisible by 8? (a) 5191 (b) 5172 (c) 5180 (d) 5176
18. 18.What is the remainder when 278 is divided by 12? (a) 1 (b) 3 (c) 2 (d) 10
19. 19.What is the units digit of 9^58? (a) 0 (b) 9 (c) 1 (d) 2
20. 20.What is the smallest number you must add to 229 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
21. 21.What is the units digit of 2^21? (a) 8 (b) 4 (c) 2 (d) 6
22. 22.What is the units digit of 9^42? (a) 1 (b) 0 (c) 9 (d) 2
23. 23.Which of these numbers is divisible by 4? (a) 1618 (b) 1609 (c) 1612 (d) 1615
24. 24.What is the smallest number you must add to 422 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
25. 25.What is the units digit of 2^37? (a) 6 (b) 2 (c) 8 (d) 4
26. 26.What is the smallest number you must add to 126 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
27. 27.What is the smallest number you must add to 323 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
28. 28.What is the units digit of 9^48? (a) 9 (b) 1 (c) 0 (d) 2
29. 29.What is the remainder when 667 is divided by 9? (a) 0 (b) 8 (c) 1 (d) 2
30. 30.What is the remainder when 70 is divided by 3? (a) 1 (b) 0 (c) 2
31. 31.What is the remainder when 197 is divided by 12? (a) 5 (b) 7 (c) 4 (d) 6
32. 32.What is the smallest number you must add to 462 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
33. 33.Which of these numbers is divisible by 8? (a) 4680 (b) 4693 (c) 4678 (d) 4682
34. 34.Which of these numbers is divisible by 9? (a) 5673 (b) 5670 (c) 5684 (d) 5662
35. 35.Which of these numbers is divisible by 9? (a) 728 (b) 715 (c) 720 (d) 734
36. 36.Which of these numbers is divisible by 9? (a) 5788 (b) 5777 (c) 5778 (d) 5781
37. 37.What is the units digit of 7^49? (a) 7 (b) 1 (c) 3 (d) 9
38. 38.What is the smallest number you must add to 519 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
39. 39.What is the smallest number you must add to 285 to make it divisible by 12? (a) 3 (b) 9 (c) 2 (d) 4
40. 40.Which of these numbers is divisible by 9? (a) 2439 (b) 2453 (c) 2436 (d) 2440

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