Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 663 · 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.Which of these numbers is divisible by 9? (a) 1306 (b) 1296 (c) 1293 (d) 1301
2. 2.What is the remainder when 137 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 6
3. 3.What is the units digit of 3^52? (a) 7 (b) 3 (c) 1 (d) 9
4. 4.What is the smallest number you must add to 307 to make it divisible by 4? (a) 2 (b) 1 (c) 3
5. 5.What is the smallest number you must add to 194 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
6. 6.What is the remainder when 51 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
7. 7.Which of these numbers is divisible by 9? (a) 7493 (b) 7479 (c) 7473 (d) 7482
8. 8.What is the remainder when 354 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
9. 9.What is the remainder when 90 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
10. 10.What is the remainder when 622 is divided by 11? (a) 6 (b) 8 (c) 5 (d) 7
11. 11.What is the units digit of 2^48? (a) 8 (b) 6 (c) 2 (d) 4
12. 12.What is the smallest number you must add to 298 to make it divisible by 12? (a) 10 (b) 3 (c) 2 (d) 1
13. 13.What is the remainder when 349 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
14. 14.Which of these numbers is divisible by 8? (a) 4959 (b) 4970 (c) 4966 (d) 4960
15. 15.What is the smallest number you must add to 723 to make it divisible by 12? (a) 10 (b) 9 (c) 8 (d) 3
16. 16.What is the remainder when 601 is divided by 11? (a) 4 (b) 8 (c) 6 (d) 7
17. 17.Which of these numbers is divisible by 3? (a) 2417 (b) 2422 (c) 2419 (d) 2418
18. 18.What is the units digit of 8^26? (a) 6 (b) 8 (c) 2 (d) 4
19. 19.Which of these numbers is divisible by 11? (a) 6196 (b) 6191 (c) 6212 (d) 6193
20. 20.What is the remainder when 698 is divided by 9? (a) 7 (b) 6 (c) 4 (d) 5
21. 21.What is the remainder when 49 is divided by 3? (a) 1 (b) 2 (c) 0
22. 22.Which of these numbers is divisible by 8? (a) 6642 (b) 6640 (c) 6650 (d) 6639
23. 23.What is the smallest number you must add to 252 to make it divisible by 10? (a) 9 (b) 2 (c) 7 (d) 8
24. 24.Which of these numbers is divisible by 6? (a) 5199 (b) 5206 (c) 5196 (d) 5194
25. 25.Which of these numbers is divisible by 6? (a) 475 (b) 471 (c) 468 (d) 467
26. 26.Which of these numbers is divisible by 11? (a) 1857 (b) 1842 (c) 1836 (d) 1837
27. 27.What is the units digit of 9^31? (a) 1 (b) 9 (c) 8 (d) 7
28. 28.Which of these numbers is divisible by 8? (a) 6052 (b) 6040 (c) 6043 (d) 6039
29. 29.What is the remainder when 340 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
30. 30.What is the smallest number you must add to 841 to make it divisible by 12? (a) 9 (b) 1 (c) 11 (d) 10
31. 31.What is the smallest number you must add to 186 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 10
32. 32.What is the smallest number you must add to 267 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
33. 33.What is the smallest number you must add to 275 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 1
34. 34.Which of these numbers is divisible by 9? (a) 6599 (b) 6610 (c) 6620 (d) 6606
35. 35.Which of these numbers is divisible by 9? (a) 4895 (b) 4896 (c) 4908 (d) 4904
36. 36.Which of these numbers is divisible by 6? (a) 822 (b) 820 (c) 825 (d) 830
37. 37.What is the units digit of 2^47? (a) 4 (b) 2 (c) 6 (d) 8
38. 38.What is the units digit of 4^21? (a) 6 (b) 4 (c) 3 (d) 5
39. 39.What is the remainder when 404 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 110 to make it divisible by 7? (a) 2 (b) 1 (c) 5 (d) 3

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