Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 3 · 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 remainder when 148 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
2. 2.What is the remainder when 237 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 6
3. 3.Which of these numbers is divisible by 3? (a) 2267 (b) 2270 (c) 2265 (d) 2264
4. 4.What is the units digit of 2^59? (a) 2 (b) 4 (c) 8 (d) 6
5. 5.Which of these numbers is divisible by 6? (a) 998 (b) 1003 (c) 1011 (d) 1002
6. 6.What is the units digit of 3^17? (a) 7 (b) 9 (c) 3 (d) 1
7. 7.What is the remainder when 270 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
8. 8.What is the smallest number you must add to 209 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the units digit of 7^59? (a) 3 (b) 1 (c) 7 (d) 9
10. 10.What is the remainder when 684 is divided by 11? (a) 2 (b) 9 (c) 3 (d) 1
11. 11.What is the remainder when 199 is divided by 10? (a) 9 (b) 7 (c) 8 (d) 1
12. 12.What is the remainder when 138 is divided by 10? (a) 2 (b) 7 (c) 8 (d) 9
13. 13.Which of these numbers is divisible by 8? (a) 4869 (b) 4853 (c) 4858 (d) 4856
14. 14.Which of these numbers is divisible by 3? (a) 1844 (b) 1847 (c) 1842 (d) 1841
15. 15.What is the remainder when 223 is divided by 3? (a) 0 (b) 1 (c) 2
16. 16.What is the remainder when 68 is divided by 3? (a) 0 (b) 1 (c) 2
17. 17.Which of these numbers is divisible by 9? (a) 3909 (b) 3906 (c) 3919 (d) 3905
18. 18.Which of these numbers is divisible by 11? (a) 2608 (b) 2598 (c) 2596 (d) 2587
19. 19.What is the units digit of 2^51? (a) 2 (b) 4 (c) 8 (d) 6
20. 20.Which of these numbers is divisible by 4? (a) 1861 (b) 1860 (c) 1867 (d) 1857
21. 21.What is the remainder when 170 is divided by 8? (a) 2 (b) 6 (c) 1 (d) 3
22. 22.What is the remainder when 161 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
23. 23.What is the remainder when 209 is divided by 3? (a) 0 (b) 1 (c) 2
24. 24.Which of these numbers is divisible by 9? (a) 7613 (b) 7596 (c) 7593 (d) 7600
25. 25.What is the units digit of 4^54? (a) 4 (b) 5 (c) 6 (d) 7
26. 26.What is the smallest number you must add to 241 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
27. 27.Which of these numbers is divisible by 8? (a) 7055 (b) 7040 (c) 7044 (d) 7036
28. 28.What is the remainder when 535 is divided by 10? (a) 5 (b) 4 (c) 7 (d) 6
29. 29.Which of these numbers is divisible by 3? (a) 418 (b) 422 (c) 417 (d) 416
30. 30.Which of these numbers is divisible by 8? (a) 1105 (b) 1100 (c) 1114 (d) 1104
31. 31.What is the smallest number you must add to 272 to make it divisible by 10? (a) 2 (b) 9 (c) 7 (d) 8
32. 32.Which of these numbers is divisible by 9? (a) 353 (b) 368 (c) 360 (d) 370
33. 33.Which of these numbers is divisible by 4? (a) 2724 (b) 2723 (c) 2727 (d) 2730
34. 34.What is the units digit of 9^60? (a) 2 (b) 0 (c) 9 (d) 1
35. 35.What is the units digit of 8^10? (a) 6 (b) 2 (c) 8 (d) 4
36. 36.What is the units digit of 8^12? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.What is the units digit of 8^20? (a) 4 (b) 2 (c) 6 (d) 8
38. 38.What is the smallest number you must add to 373 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 6
39. 39.What is the units digit of 7^38? (a) 3 (b) 7 (c) 1 (d) 9
40. 40.Which of these numbers is divisible by 4? (a) 2040 (b) 2038 (c) 2042 (d) 2047

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