Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 272 · 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 116 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 2
2. 2.What is the smallest number you must add to 65 to make it divisible by 4? (a) 3 (b) 2 (c) 1
3. 3.What is the smallest number you must add to 67 to make it divisible by 3? (a) 2 (b) 1
4. 4.Which of these numbers is divisible by 9? (a) 5665 (b) 5644 (c) 5654 (d) 5652
5. 5.Which of these numbers is divisible by 6? (a) 4878 (b) 4877 (c) 4888 (d) 4880
6. 6.What is the units digit of 4^57? (a) 4 (b) 3 (c) 6 (d) 5
7. 7.Which of these numbers is divisible by 6? (a) 1315 (b) 1322 (c) 1327 (d) 1320
8. 8.What is the remainder when 379 is divided by 12? (a) 7 (b) 6 (c) 5 (d) 8
9. 9.What is the remainder when 251 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
10. 10.Which of these numbers is divisible by 9? (a) 3049 (b) 3042 (c) 3057 (d) 3035
11. 11.What is the units digit of 8^48? (a) 8 (b) 6 (c) 4 (d) 2
12. 12.What is the smallest number you must add to 53 to make it divisible by 4? (a) 2 (b) 3 (c) 1
13. 13.What is the units digit of 7^49? (a) 9 (b) 3 (c) 7 (d) 1
14. 14.Which of these numbers is divisible by 6? (a) 1045 (b) 1053 (c) 1043 (d) 1044
15. 15.What is the units digit of 9^19? (a) 9 (b) 8 (c) 1 (d) 7
16. 16.Which of these numbers is divisible by 8? (a) 433 (b) 418 (c) 427 (d) 424
17. 17.What is the smallest number you must add to 582 to make it divisible by 11? (a) 2 (b) 3 (c) 10 (d) 1
18. 18.Which of these numbers is divisible by 3? (a) 1634 (b) 1629 (c) 1627 (d) 1630
19. 19.What is the smallest number you must add to 401 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
20. 20.What is the units digit of 9^46? (a) 1 (b) 2 (c) 0 (d) 9
21. 21.What is the remainder when 161 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
22. 22.What is the units digit of 2^24? (a) 8 (b) 4 (c) 2 (d) 6
23. 23.What is the smallest number you must add to 663 to make it divisible by 10? (a) 8 (b) 3 (c) 6 (d) 7
24. 24.What is the smallest number you must add to 77 to make it divisible by 4? (a) 2 (b) 3 (c) 1
25. 25.What is the units digit of 2^27? (a) 4 (b) 8 (c) 2 (d) 6
26. 26.What is the smallest number you must add to 200 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
27. 27.What is the remainder when 89 is divided by 7? (a) 2 (b) 5 (c) 6 (d) 4
28. 28.What is the smallest number you must add to 309 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 1
29. 29.Which of these numbers is divisible by 6? (a) 316 (b) 301 (c) 306 (d) 307
30. 30.Which of these numbers is divisible by 8? (a) 1303 (b) 1290 (c) 1282 (d) 1288
31. 31.Which of these numbers is divisible by 3? (a) 1618 (b) 1617 (c) 1621 (d) 1616
32. 32.Which of these numbers is divisible by 6? (a) 4996 (b) 4982 (c) 4989 (d) 4986
33. 33.What is the units digit of 3^15? (a) 3 (b) 9 (c) 7 (d) 1
34. 34.Which of these numbers is divisible by 4? (a) 1504 (b) 1505 (c) 1501 (d) 1510
35. 35.What is the smallest number you must add to 237 to make it divisible by 9? (a) 3 (b) 5 (c) 7 (d) 6
36. 36.Which of these numbers is divisible by 3? (a) 1834 (b) 1831 (c) 1837 (d) 1833
37. 37.What is the remainder when 298 is divided by 8? (a) 2 (b) 6 (c) 1 (d) 3
38. 38.What is the remainder when 101 is divided by 3? (a) 1 (b) 2 (c) 0
39. 39.What is the remainder when 559 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
40. 40.What is the units digit of 8^30? (a) 4 (b) 8 (c) 2 (d) 6

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