Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 225 · 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 9^45? (a) 8 (b) 9 (c) 1 (d) 7
2. 2.What is the units digit of 7^19? (a) 9 (b) 1 (c) 7 (d) 3
3. 3.What is the smallest number you must add to 368 to make it divisible by 12? (a) 8 (b) 5 (c) 4 (d) 3
4. 4.Which of these numbers is divisible by 9? (a) 2736 (b) 2750 (c) 2730 (d) 2740
5. 5.What is the remainder when 90 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
6. 6.What is the units digit of 2^39? (a) 4 (b) 2 (c) 6 (d) 8
7. 7.Which of these numbers is divisible by 9? (a) 6035 (b) 6039 (c) 6055 (d) 6041
8. 8.What is the units digit of 7^35? (a) 7 (b) 9 (c) 1 (d) 3
9. 9.What is the remainder when 537 is divided by 12? (a) 9 (b) 8 (c) 3 (d) 10
10. 10.What is the remainder when 182 is divided by 3? (a) 2 (b) 1 (c) 0
11. 11.Which of these numbers is divisible by 3? (a) 1724 (b) 1718 (c) 1719 (d) 1721
12. 12.Which of these numbers is divisible by 3? (a) 1117 (b) 1120 (c) 1119 (d) 1123
13. 13.What is the remainder when 359 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
14. 14.Which of these numbers is divisible by 3? (a) 204 (b) 203 (c) 209 (d) 205
15. 15.What is the remainder when 329 is divided by 6? (a) 1 (b) 5 (c) 4 (d) 3
16. 16.Which of these numbers is divisible by 8? (a) 3417 (b) 3416 (c) 3413 (d) 3430
17. 17.What is the remainder when 128 is divided by 3? (a) 0 (b) 2 (c) 1
18. 18.Which of these numbers is divisible by 6? (a) 3610 (b) 3597 (c) 3600 (d) 3601
19. 19.What is the remainder when 50 is divided by 3? (a) 1 (b) 2 (c) 0
20. 20.Which of these numbers is divisible by 9? (a) 7044 (b) 7037 (c) 7029 (d) 7023
21. 21.What is the smallest number you must add to 359 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3
22. 22.What is the units digit of 4^47? (a) 5 (b) 4 (c) 6 (d) 3
23. 23.What is the remainder when 860 is divided by 12? (a) 7 (b) 4 (c) 8 (d) 9
24. 24.What is the units digit of 8^12? (a) 4 (b) 6 (c) 2 (d) 8
25. 25.What is the smallest number you must add to 206 to make it divisible by 4? (a) 1 (b) 2 (c) 3
26. 26.Which of these numbers is divisible by 11? (a) 509 (b) 517 (c) 530 (d) 519
27. 27.What is the units digit of 3^30? (a) 7 (b) 3 (c) 1 (d) 9
28. 28.What is the remainder when 32 is divided by 3? (a) 0 (b) 2 (c) 1
29. 29.What is the units digit of 9^19? (a) 7 (b) 1 (c) 9 (d) 8
30. 30.What is the remainder when 504 is divided by 11? (a) 8 (b) 10 (c) 9 (d) 2
31. 31.What is the units digit of 9^20? (a) 1 (b) 0 (c) 2 (d) 9
32. 32.What is the smallest number you must add to 431 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 5
33. 33.What is the smallest number you must add to 209 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 9
34. 34.What is the remainder when 170 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
35. 35.What is the smallest number you must add to 356 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
36. 36.What is the remainder when 222 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 2
37. 37.Which of these numbers is divisible by 6? (a) 5245 (b) 5253 (c) 5244 (d) 5241
38. 38.What is the remainder when 133 is divided by 3? (a) 1 (b) 2 (c) 0
39. 39.What is the remainder when 399 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
40. 40.Which of these numbers is divisible by 3? (a) 1868 (b) 1869 (c) 1874 (d) 1871

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