Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 307 · 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 smallest number you must add to 827 to make it divisible by 11? (a) 2 (b) 8 (c) 10 (d) 9
2. 2.What is the remainder when 142 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
3. 3.Which of these numbers is divisible by 6? (a) 4221 (b) 4211 (c) 4212 (d) 4214
4. 4.What is the smallest number you must add to 391 to make it divisible by 9? (a) 5 (b) 7 (c) 6 (d) 4
5. 5.What is the units digit of 2^11? (a) 8 (b) 2 (c) 6 (d) 4
6. 6.What is the smallest number you must add to 325 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 3
7. 7.What is the units digit of 8^30? (a) 4 (b) 6 (c) 2 (d) 8
8. 8.What is the remainder when 422 is divided by 12? (a) 2 (b) 1 (c) 3 (d) 10
9. 9.What is the units digit of 7^37? (a) 1 (b) 3 (c) 7 (d) 9
10. 10.What is the smallest number you must add to 652 to make it divisible by 12? (a) 4 (b) 7 (c) 8 (d) 9
11. 11.What is the smallest number you must add to 510 to make it divisible by 9? (a) 3 (b) 6 (c) 2 (d) 4
12. 12.Which of these numbers is divisible by 8? (a) 1859 (b) 1875 (c) 1864 (d) 1866
13. 13.What is the units digit of 4^20? (a) 5 (b) 7 (c) 6 (d) 4
14. 14.What is the smallest number you must add to 187 to make it divisible by 3? (a) 1 (b) 2
15. 15.Which of these numbers is divisible by 11? (a) 7898 (b) 7894 (c) 7907 (d) 7919
16. 16.Which of these numbers is divisible by 8? (a) 5464 (b) 5461 (c) 5475 (d) 5470
17. 17.What is the remainder when 200 is divided by 9? (a) 1 (b) 2 (c) 7 (d) 3
18. 18.What is the smallest number you must add to 345 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
19. 19.What is the smallest number you must add to 102 to make it divisible by 4? (a) 3 (b) 1 (c) 2
20. 20.What is the units digit of 2^35? (a) 2 (b) 6 (c) 4 (d) 8
21. 21.Which of these numbers is divisible by 9? (a) 7537 (b) 7544 (c) 7530 (d) 7533
22. 22.What is the smallest number you must add to 113 to make it divisible by 4? (a) 2 (b) 1 (c) 3
23. 23.What is the remainder when 440 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
24. 24.Which of these numbers is divisible by 9? (a) 7687 (b) 7679 (c) 7677 (d) 7673
25. 25.What is the units digit of 3^13? (a) 3 (b) 9 (c) 7 (d) 1
26. 26.What is the units digit of 9^54? (a) 1 (b) 9 (c) 2 (d) 0
27. 27.What is the units digit of 4^60? (a) 6 (b) 4 (c) 7 (d) 5
28. 28.Which of these numbers is divisible by 11? (a) 3117 (b) 3133 (c) 3112 (d) 3113
29. 29.What is the units digit of 9^13? (a) 9 (b) 7 (c) 1 (d) 8
30. 30.Which of these numbers is divisible by 9? (a) 720 (b) 714 (c) 736 (d) 724
31. 31.What is the smallest number you must add to 493 to make it divisible by 7? (a) 6 (b) 5 (c) 3 (d) 4
32. 32.Which of these numbers is divisible by 11? (a) 1832 (b) 1819 (c) 1815 (d) 1805
33. 33.Which of these numbers is divisible by 11? (a) 2904 (b) 2909 (c) 2925 (d) 2902
34. 34.What is the smallest number you must add to 422 to make it divisible by 9? (a) 1 (b) 3 (c) 8 (d) 2
35. 35.Which of these numbers is divisible by 4? (a) 1954 (b) 1951 (c) 1952 (d) 1957
36. 36.What is the remainder when 110 is divided by 3? (a) 0 (b) 2 (c) 1
37. 37.Which of these numbers is divisible by 9? (a) 7915 (b) 7937 (c) 7923 (d) 7920
38. 38.What is the units digit of 9^47? (a) 8 (b) 7 (c) 1 (d) 9
39. 39.What is the smallest number you must add to 68 to make it divisible by 3? (a) 1 (b) 2
40. 40.Which of these numbers is divisible by 11? (a) 9889 (b) 9902 (c) 9899 (d) 9881

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