Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 631 · 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 4? (a) 1072 (b) 1070 (c) 1073 (d) 1079
2. 2.Which of these numbers is divisible by 3? (a) 1885 (b) 1882 (c) 1888 (d) 1884
3. 3.What is the remainder when 106 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
4. 4.What is the units digit of 8^29? (a) 2 (b) 8 (c) 6 (d) 4
5. 5.Which of these numbers is divisible by 8? (a) 2532 (b) 2518 (c) 2520 (d) 2524
6. 6.Which of these numbers is divisible by 3? (a) 316 (b) 320 (c) 314 (d) 315
7. 7.Which of these numbers is divisible by 8? (a) 5951 (b) 5939 (c) 5955 (d) 5944
8. 8.Which of these numbers is divisible by 6? (a) 3257 (b) 3246 (c) 3245 (d) 3251
9. 9.What is the units digit of 9^57? (a) 7 (b) 8 (c) 9 (d) 1
10. 10.What is the units digit of 3^10? (a) 7 (b) 3 (c) 9 (d) 1
11. 11.What is the units digit of 9^25? (a) 9 (b) 8 (c) 7 (d) 1
12. 12.Which of these numbers is divisible by 3? (a) 1942 (b) 1945 (c) 1941 (d) 1940
13. 13.What is the smallest number you must add to 348 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
14. 14.What is the smallest number you must add to 434 to make it divisible by 12? (a) 11 (b) 10 (c) 2 (d) 9
15. 15.What is the smallest number you must add to 118 to make it divisible by 4? (a) 2 (b) 3 (c) 1
16. 16.What is the units digit of 3^18? (a) 3 (b) 9 (c) 7 (d) 1
17. 17.Which of these numbers is divisible by 3? (a) 2137 (b) 2133 (c) 2135 (d) 2132
18. 18.What is the smallest number you must add to 314 to make it divisible by 9? (a) 1 (b) 8 (c) 3 (d) 2
19. 19.What is the remainder when 197 is divided by 7? (a) 1 (b) 2 (c) 6 (d) 0
20. 20.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
21. 21.What is the smallest number you must add to 520 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
22. 22.What is the smallest number you must add to 251 to make it divisible by 9? (a) 8 (b) 2 (c) 1 (d) 3
23. 23.What is the remainder when 786 is divided by 11? (a) 7 (b) 5 (c) 4 (d) 6
24. 24.What is the units digit of 3^54? (a) 9 (b) 3 (c) 7 (d) 1
25. 25.What is the smallest number you must add to 263 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
26. 26.What is the units digit of 7^60? (a) 7 (b) 9 (c) 3 (d) 1
27. 27.What is the remainder when 214 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
28. 28.Which of these numbers is divisible by 3? (a) 185 (b) 181 (c) 179 (d) 180
29. 29.Which of these numbers is divisible by 8? (a) 899 (b) 914 (c) 906 (d) 904
30. 30.Which of these numbers is divisible by 9? (a) 7726 (b) 7722 (c) 7721 (d) 7736
31. 31.Which of these numbers is divisible by 11? (a) 5448 (b) 5468 (c) 5456 (d) 5462
32. 32.What is the smallest number you must add to 413 to make it divisible by 10? (a) 6 (b) 7 (c) 8 (d) 3
33. 33.What is the units digit of 2^16? (a) 8 (b) 6 (c) 2 (d) 4
34. 34.Which of these numbers is divisible by 8? (a) 5262 (b) 5256 (c) 5269 (d) 5253
35. 35.What is the remainder when 683 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1
36. 36.Which of these numbers is divisible by 6? (a) 2458 (b) 2460 (c) 2468 (d) 2461
37. 37.What is the units digit of 7^58? (a) 7 (b) 9 (c) 1 (d) 3
38. 38.What is the remainder when 200 is divided by 9? (a) 1 (b) 2 (c) 3 (d) 7
39. 39.Which of these numbers is divisible by 6? (a) 4366 (b) 4353 (c) 4356 (d) 4361
40. 40.What is the smallest number you must add to 645 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 2

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