Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 886 · 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 500 to make it divisible by 12? (a) 8 (b) 4 (c) 3 (d) 5
2. 2.What is the smallest number you must add to 147 to make it divisible by 10? (a) 7 (b) 3 (c) 4 (d) 2
3. 3.What is the smallest number you must add to 602 to make it divisible by 9? (a) 1 (b) 8 (c) 2 (d) 3
4. 4.Which of these numbers is divisible by 6? (a) 1359 (b) 1363 (c) 1356 (d) 1355
5. 5.Which of these numbers is divisible by 4? (a) 2037 (b) 2036 (c) 2035 (d) 2041
6. 6.What is the remainder when 297 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
7. 7.What is the remainder when 344 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
8. 8.What is the units digit of 8^56? (a) 4 (b) 6 (c) 8 (d) 2
9. 9.What is the smallest number you must add to 100 to make it divisible by 3? (a) 2 (b) 1
10. 10.What is the remainder when 183 is divided by 10? (a) 3 (b) 7 (c) 2 (d) 4
11. 11.What is the smallest number you must add to 513 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
12. 12.What is the remainder when 267 is divided by 10? (a) 8 (b) 3 (c) 6 (d) 7
13. 13.What is the remainder when 356 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
14. 14.What is the remainder when 179 is divided by 11? (a) 2 (b) 8 (c) 3 (d) 4
15. 15.Which of these numbers is divisible by 3? (a) 1761 (b) 1760 (c) 1762 (d) 1766
16. 16.What is the remainder when 506 is divided by 9? (a) 1 (b) 3 (c) 7 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 3438 (b) 3436 (c) 3435 (d) 3441
18. 18.Which of these numbers is divisible by 4? (a) 639 (b) 635 (c) 643 (d) 636
19. 19.Which of these numbers is divisible by 11? (a) 1545 (b) 1554 (c) 1539 (d) 1540
20. 20.Which of these numbers is divisible by 3? (a) 1797 (b) 1801 (c) 1798 (d) 1796
21. 21.What is the smallest number you must add to 68 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
22. 22.What is the remainder when 178 is divided by 3? (a) 0 (b) 2 (c) 1
23. 23.What is the remainder when 242 is divided by 3? (a) 0 (b) 1 (c) 2
24. 24.Which of these numbers is divisible by 9? (a) 2508 (b) 2511 (c) 2519 (d) 2521
25. 25.What is the units digit of 7^54? (a) 1 (b) 7 (c) 3 (d) 9
26. 26.What is the remainder when 700 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 2
27. 27.What is the units digit of 3^24? (a) 9 (b) 3 (c) 7 (d) 1
28. 28.What is the smallest number you must add to 223 to make it divisible by 12? (a) 4 (b) 5 (c) 6 (d) 7
29. 29.What is the remainder when 47 is divided by 3? (a) 1 (b) 2 (c) 0
30. 30.What is the smallest number you must add to 412 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
31. 31.What is the units digit of 9^33? (a) 1 (b) 8 (c) 7 (d) 9
32. 32.Which of these numbers is divisible by 11? (a) 1871 (b) 1869 (c) 1856 (d) 1859
33. 33.What is the remainder when 300 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 1
34. 34.What is the units digit of 3^21? (a) 1 (b) 3 (c) 7 (d) 9
35. 35.What is the remainder when 668 is divided by 12? (a) 4 (b) 9 (c) 7 (d) 8
36. 36.Which of these numbers is divisible by 11? (a) 5118 (b) 5128 (c) 5115 (d) 5108
37. 37.What is the units digit of 3^12? (a) 7 (b) 1 (c) 3 (d) 9
38. 38.Which of these numbers is divisible by 4? (a) 1464 (b) 1469 (c) 1467 (d) 1461
39. 39.What is the remainder when 213 is divided by 12? (a) 10 (b) 9 (c) 3 (d) 8
40. 40.What is the units digit of 2^25? (a) 6 (b) 8 (c) 4 (d) 2

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