Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 519 · 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 units digit of 4^21? (a) 6 (b) 4 (c) 5 (d) 3
2. 2.What is the remainder when 44 is divided by 3? (a) 1 (b) 2 (c) 0
3. 3.What is the units digit of 4^20? (a) 4 (b) 5 (c) 7 (d) 6
4. 4.What is the smallest number you must add to 893 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
5. 5.Which of these numbers is divisible by 11? (a) 5890 (b) 5896 (c) 5902 (d) 5912
6. 6.What is the units digit of 3^55? (a) 7 (b) 9 (c) 1 (d) 3
7. 7.Which of these numbers is divisible by 4? (a) 355 (b) 356 (c) 359 (d) 361
8. 8.Which of these numbers is divisible by 11? (a) 5082 (b) 5080 (c) 5096 (d) 5084
9. 9.What is the smallest number you must add to 45 to make it divisible by 4? (a) 1 (b) 3 (c) 2
10. 10.What is the smallest number you must add to 302 to make it divisible by 4? (a) 3 (b) 2 (c) 1
11. 11.Which of these numbers is divisible by 11? (a) 2664 (b) 2662 (c) 2654 (d) 2680
12. 12.What is the remainder when 187 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
13. 13.Which of these numbers is divisible by 9? (a) 6597 (b) 6591 (c) 6614 (d) 6601
14. 14.Which of these numbers is divisible by 4? (a) 1908 (b) 1914 (c) 1910 (d) 1906
15. 15.What is the remainder when 613 is divided by 10? (a) 3 (b) 7 (c) 2 (d) 4
16. 16.Which of these numbers is divisible by 9? (a) 2617 (b) 2632 (c) 2621 (d) 2619
17. 17.What is the smallest number you must add to 391 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
18. 18.What is the remainder when 889 is divided by 12? (a) 0 (b) 2 (c) 1 (d) 11
19. 19.What is the smallest number you must add to 573 to make it divisible by 8? (a) 4 (b) 2 (c) 3 (d) 5
20. 20.What is the remainder when 597 is divided by 12? (a) 8 (b) 9 (c) 10 (d) 3
21. 21.Which of these numbers is divisible by 6? (a) 4805 (b) 4796 (c) 4790 (d) 4794
22. 22.What is the units digit of 9^43? (a) 1 (b) 8 (c) 9 (d) 7
23. 23.What is the smallest number you must add to 91 to make it divisible by 4? (a) 2 (b) 1 (c) 3
24. 24.What is the smallest number you must add to 178 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 2538 (b) 2540 (c) 2537 (d) 2543
26. 26.What is the remainder when 659 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 1
27. 27.Which of these numbers is divisible by 3? (a) 1166 (b) 1162 (c) 1164 (d) 1169
28. 28.What is the units digit of 8^24? (a) 8 (b) 6 (c) 4 (d) 2
29. 29.What is the smallest number you must add to 166 to make it divisible by 3? (a) 2 (b) 1
30. 30.What is the smallest number you must add to 941 to make it divisible by 12? (a) 5 (b) 8 (c) 7 (d) 6
31. 31.Which of these numbers is divisible by 6? (a) 2673 (b) 2668 (c) 2662 (d) 2664
32. 32.What is the remainder when 133 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
33. 33.What is the smallest number you must add to 152 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
34. 34.What is the remainder when 586 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
35. 35.Which of these numbers is divisible by 9? (a) 2208 (b) 2214 (c) 2225 (d) 2217
36. 36.What is the units digit of 8^55? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.What is the units digit of 3^18? (a) 3 (b) 7 (c) 9 (d) 1
38. 38.Which of these numbers is divisible by 9? (a) 5195 (b) 5180 (c) 5184 (d) 5186
39. 39.Which of these numbers is divisible by 4? (a) 2469 (b) 2468 (c) 2474 (d) 2467
40. 40.What is the units digit of 8^51? (a) 6 (b) 8 (c) 2 (d) 4

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