Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 732 · 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) 1734 (b) 1720 (c) 1714 (d) 1719
2. 2.What is the remainder when 117 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
3. 3.What is the units digit of 3^25? (a) 9 (b) 3 (c) 7 (d) 1
4. 4.Which of these numbers is divisible by 4? (a) 2742 (b) 2747 (c) 2744 (d) 2749
5. 5.What is the units digit of 8^49? (a) 6 (b) 2 (c) 8 (d) 4
6. 6.What is the smallest number you must add to 145 to make it divisible by 8? (a) 6 (b) 7 (c) 1 (d) 5
7. 7.What is the smallest number you must add to 299 to make it divisible by 4? (a) 2 (b) 1 (c) 3
8. 8.What is the smallest number you must add to 794 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 2
9. 9.What is the units digit of 8^59? (a) 4 (b) 2 (c) 8 (d) 6
10. 10.What is the smallest number you must add to 945 to make it divisible by 12? (a) 9 (b) 3 (c) 4 (d) 2
11. 11.What is the units digit of 3^27? (a) 7 (b) 1 (c) 3 (d) 9
12. 12.Which of these numbers is divisible by 6? (a) 723 (b) 728 (c) 726 (d) 737
13. 13.What is the units digit of 2^54? (a) 4 (b) 8 (c) 2 (d) 6
14. 14.What is the units digit of 9^29? (a) 1 (b) 7 (c) 9 (d) 8
15. 15.What is the units digit of 8^18? (a) 2 (b) 8 (c) 4 (d) 6
16. 16.What is the remainder when 528 is divided by 10? (a) 8 (b) 2 (c) 7 (d) 9
17. 17.What is the smallest number you must add to 281 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
18. 18.What is the remainder when 384 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
19. 19.What is the remainder when 286 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
20. 20.What is the remainder when 376 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
21. 21.What is the units digit of 8^19? (a) 2 (b) 4 (c) 6 (d) 8
22. 22.What is the remainder when 303 is divided by 12? (a) 4 (b) 9 (c) 2 (d) 3
23. 23.What is the units digit of 8^44? (a) 2 (b) 4 (c) 8 (d) 6
24. 24.What is the smallest number you must add to 179 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
25. 25.What is the remainder when 77 is divided by 3? (a) 2 (b) 0 (c) 1
26. 26.What is the remainder when 157 is divided by 3? (a) 0 (b) 2 (c) 1
27. 27.What is the smallest number you must add to 441 to make it divisible by 10? (a) 8 (b) 7 (c) 1 (d) 9
28. 28.What is the remainder when 81 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
29. 29.What is the units digit of 2^17? (a) 6 (b) 8 (c) 2 (d) 4
30. 30.What is the smallest number you must add to 774 to make it divisible by 11? (a) 4 (b) 8 (c) 7 (d) 6
31. 31.What is the units digit of 9^57? (a) 1 (b) 7 (c) 8 (d) 9
32. 32.What is the smallest number you must add to 109 to make it divisible by 4? (a) 3 (b) 2 (c) 1
33. 33.What is the units digit of 3^43? (a) 3 (b) 1 (c) 7 (d) 9
34. 34.Which of these numbers is divisible by 11? (a) 2537 (b) 2517 (c) 2519 (d) 2524
35. 35.What is the units digit of 9^30? (a) 0 (b) 1 (c) 9 (d) 2
36. 36.What is the smallest number you must add to 237 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2
37. 37.Which of these numbers is divisible by 4? (a) 1897 (b) 1907 (c) 1901 (d) 1900
38. 38.Which of these numbers is divisible by 11? (a) 6477 (b) 6479 (c) 6498 (d) 6488
39. 39.What is the smallest number you must add to 34 to make it divisible by 3? (a) 2 (b) 1
40. 40.What is the remainder when 250 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1

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