Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 157 · 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 204 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 7
2. 2.What is the remainder when 62 is divided by 3? (a) 1 (b) 0 (c) 2
3. 3.What is the smallest number you must add to 149 to make it divisible by 4? (a) 1 (b) 2 (c) 3
4. 4.Which of these numbers is divisible by 8? (a) 1799 (b) 1792 (c) 1805 (d) 1787
5. 5.What is the units digit of 3^52? (a) 9 (b) 7 (c) 1 (d) 3
6. 6.What is the units digit of 7^30? (a) 7 (b) 1 (c) 9 (d) 3
7. 7.Which of these numbers is divisible by 3? (a) 456 (b) 458 (c) 460 (d) 455
8. 8.What is the remainder when 289 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
9. 9.What is the smallest number you must add to 334 to make it divisible by 8? (a) 3 (b) 6 (c) 2 (d) 1
10. 10.What is the smallest number you must add to 92 to make it divisible by 8? (a) 5 (b) 6 (c) 3 (d) 4
11. 11.Which of these numbers is divisible by 3? (a) 883 (b) 882 (c) 886 (d) 881
12. 12.What is the units digit of 3^32? (a) 1 (b) 3 (c) 9 (d) 7
13. 13.What is the smallest number you must add to 231 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
14. 14.Which of these numbers is divisible by 4? (a) 521 (b) 525 (c) 529 (d) 524
15. 15.Which of these numbers is divisible by 6? (a) 2100 (b) 2098 (c) 2103 (d) 2110
16. 16.What is the remainder when 409 is divided by 8? (a) 1 (b) 0 (c) 2 (d) 7
17. 17.Which of these numbers is divisible by 3? (a) 989 (b) 990 (c) 994 (d) 991
18. 18.What is the smallest number you must add to 187 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 5
19. 19.Which of these numbers is divisible by 6? (a) 1476 (b) 1475 (c) 1486 (d) 1481
20. 20.Which of these numbers is divisible by 3? (a) 2132 (b) 2129 (c) 2130 (d) 2135
21. 21.What is the smallest number you must add to 203 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
22. 22.What is the smallest number you must add to 154 to make it divisible by 3? (a) 2 (b) 1
23. 23.What is the units digit of 3^30? (a) 3 (b) 9 (c) 7 (d) 1
24. 24.What is the smallest number you must add to 263 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 7
25. 25.Which of these numbers is divisible by 11? (a) 7109 (b) 7130 (c) 7117 (d) 7127
26. 26.What is the remainder when 194 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 2
27. 27.What is the remainder when 171 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
28. 28.Which of these numbers is divisible by 4? (a) 975 (b) 979 (c) 981 (d) 976
29. 29.Which of these numbers is divisible by 8? (a) 1171 (b) 1168 (c) 1180 (d) 1166
30. 30.What is the smallest number you must add to 284 to make it divisible by 11? (a) 3 (b) 2 (c) 9 (d) 1
31. 31.What is the units digit of 7^53? (a) 3 (b) 7 (c) 9 (d) 1
32. 32.What is the units digit of 3^41? (a) 7 (b) 9 (c) 3 (d) 1
33. 33.Which of these numbers is divisible by 3? (a) 2454 (b) 2455 (c) 2452 (d) 2458
34. 34.What is the remainder when 43 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
35. 35.Which of these numbers is divisible by 9? (a) 3187 (b) 3174 (c) 3178 (d) 3177
36. 36.What is the remainder when 438 is divided by 8? (a) 2 (b) 6 (c) 7 (d) 5
37. 37.What is the units digit of 7^35? (a) 3 (b) 7 (c) 1 (d) 9
38. 38.What is the units digit of 2^35? (a) 8 (b) 4 (c) 2 (d) 6
39. 39.What is the smallest number you must add to 187 to make it divisible by 4? (a) 3 (b) 1 (c) 2
40. 40.What is the remainder when 703 is divided by 12? (a) 7 (b) 8 (c) 5 (d) 6

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad