Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 872 · 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 3? (a) 1506 (b) 1508 (c) 1510 (d) 1504
2. 2.What is the remainder when 358 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
3. 3.What is the smallest number you must add to 214 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
4. 4.What is the smallest number you must add to 635 to make it divisible by 11? (a) 3 (b) 2 (c) 8 (d) 4
5. 5.What is the smallest number you must add to 210 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 3
6. 6.Which of these numbers is divisible by 6? (a) 4032 (b) 4042 (c) 4036 (d) 4028
7. 7.Which of these numbers is divisible by 4? (a) 842 (b) 833 (c) 836 (d) 838
8. 8.Which of these numbers is divisible by 9? (a) 7001 (b) 7007 (c) 7015 (d) 7002
9. 9.What is the smallest number you must add to 101 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.What is the remainder when 204 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
11. 11.What is the remainder when 317 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
12. 12.What is the units digit of 3^49? (a) 7 (b) 3 (c) 1 (d) 9
13. 13.Which of these numbers is divisible by 4? (a) 449 (b) 453 (c) 448 (d) 447
14. 14.Which of these numbers is divisible by 3? (a) 939 (b) 941 (c) 938 (d) 944
15. 15.Which of these numbers is divisible by 6? (a) 456 (b) 451 (c) 460 (d) 467
16. 16.What is the units digit of 2^43? (a) 4 (b) 6 (c) 8 (d) 2
17. 17.What is the remainder when 265 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
18. 18.What is the remainder when 222 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
19. 19.What is the smallest number you must add to 103 to make it divisible by 10? (a) 7 (b) 6 (c) 3 (d) 8
20. 20.What is the remainder when 71 is divided by 5? (a) 1 (b) 0 (c) 4 (d) 2
21. 21.What is the smallest number you must add to 82 to make it divisible by 4? (a) 3 (b) 2 (c) 1
22. 22.What is the smallest number you must add to 512 to make it divisible by 11? (a) 7 (b) 6 (c) 4 (d) 5
23. 23.Which of these numbers is divisible by 3? (a) 2288 (b) 2289 (c) 2291 (d) 2294
24. 24.What is the units digit of 2^28? (a) 2 (b) 4 (c) 8 (d) 6
25. 25.What is the units digit of 4^59? (a) 3 (b) 4 (c) 6 (d) 5
26. 26.What is the units digit of 8^20? (a) 6 (b) 2 (c) 8 (d) 4
27. 27.What is the smallest number you must add to 52 to make it divisible by 3? (a) 1 (b) 2
28. 28.What is the smallest number you must add to 153 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
29. 29.What is the units digit of 4^43? (a) 3 (b) 6 (c) 5 (d) 4
30. 30.What is the smallest number you must add to 651 to make it divisible by 9? (a) 3 (b) 6 (c) 7 (d) 5
31. 31.Which of these numbers is divisible by 4? (a) 758 (b) 753 (c) 762 (d) 756
32. 32.Which of these numbers is divisible by 9? (a) 1050 (b) 1058 (c) 1068 (d) 1053
33. 33.What is the remainder when 337 is divided by 8? (a) 7 (b) 1 (c) 0 (d) 2
34. 34.What is the remainder when 125 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
35. 35.What is the remainder when 92 is divided by 3? (a) 1 (b) 0 (c) 2
36. 36.What is the remainder when 532 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
37. 37.What is the smallest number you must add to 40 to make it divisible by 3? (a) 1 (b) 2
38. 38.What is the remainder when 425 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
39. 39.What is the units digit of 2^21? (a) 2 (b) 4 (c) 6 (d) 8
40. 40.What is the smallest number you must add to 316 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6

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