Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 206 · 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) 485 (b) 481 (c) 483 (d) 487
2. 2.Which of these numbers is divisible by 4? (a) 3275 (b) 3283 (c) 3277 (d) 3276
3. 3.Which of these numbers is divisible by 11? (a) 3147 (b) 3158 (c) 3174 (d) 3157
4. 4.Which of these numbers is divisible by 3? (a) 2691 (b) 2693 (c) 2690 (d) 2695
5. 5.Which of these numbers is divisible by 4? (a) 1186 (b) 1180 (c) 1179 (d) 1182
6. 6.Which of these numbers is divisible by 4? (a) 446 (b) 455 (c) 450 (d) 448
7. 7.Which of these numbers is divisible by 11? (a) 9687 (b) 9674 (c) 9680 (d) 9698
8. 8.What is the smallest number you must add to 185 to make it divisible by 4? (a) 1 (b) 3 (c) 2
9. 9.What is the smallest number you must add to 130 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.What is the smallest number you must add to 151 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the smallest number you must add to 396 to make it divisible by 10? (a) 3 (b) 6 (c) 4 (d) 5
12. 12.What is the remainder when 84 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
13. 13.What is the smallest number you must add to 79 to make it divisible by 3? (a) 1 (b) 2
14. 14.What is the units digit of 9^10? (a) 0 (b) 2 (c) 9 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 1176 (b) 1181 (c) 1178 (d) 1174
16. 16.What is the units digit of 4^13? (a) 5 (b) 4 (c) 3 (d) 6
17. 17.Which of these numbers is divisible by 3? (a) 2342 (b) 2340 (c) 2339 (d) 2345
18. 18.Which of these numbers is divisible by 4? (a) 1166 (b) 1164 (c) 1169 (d) 1161
19. 19.What is the units digit of 9^17? (a) 1 (b) 9 (c) 7 (d) 8
20. 20.What is the smallest number you must add to 971 to make it divisible by 12? (a) 3 (b) 1 (c) 11 (d) 2
21. 21.Which of these numbers is divisible by 6? (a) 4609 (b) 4602 (c) 4600 (d) 4605
22. 22.Which of these numbers is divisible by 3? (a) 2145 (b) 2143 (c) 2147 (d) 2149
23. 23.What is the units digit of 8^40? (a) 8 (b) 2 (c) 4 (d) 6
24. 24.What is the smallest number you must add to 324 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
25. 25.What is the smallest number you must add to 643 to make it divisible by 9? (a) 7 (b) 6 (c) 5 (d) 4
26. 26.What is the remainder when 275 is divided by 8? (a) 2 (b) 3 (c) 4 (d) 5
27. 27.Which of these numbers is divisible by 9? (a) 7288 (b) 7281 (c) 7279 (d) 7291
28. 28.Which of these numbers is divisible by 9? (a) 6707 (b) 6696 (c) 6695 (d) 6702
29. 29.What is the smallest number you must add to 203 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
30. 30.Which of these numbers is divisible by 6? (a) 1181 (b) 1187 (c) 1172 (d) 1176
31. 31.Which of these numbers is divisible by 11? (a) 9035 (b) 9026 (c) 9044 (d) 9031
32. 32.What is the smallest number you must add to 101 to make it divisible by 3? (a) 2 (b) 1
33. 33.Which of these numbers is divisible by 3? (a) 656 (b) 651 (c) 652 (d) 649
34. 34.What is the remainder when 475 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
35. 35.Which of these numbers is divisible by 4? (a) 3539 (b) 3541 (c) 3545 (d) 3540
36. 36.What is the smallest number you must add to 772 to make it divisible by 12? (a) 4 (b) 7 (c) 9 (d) 8
37. 37.What is the units digit of 2^24? (a) 2 (b) 4 (c) 6 (d) 8
38. 38.Which of these numbers is divisible by 11? (a) 8461 (b) 8447 (c) 8448 (d) 8451
39. 39.Which of these numbers is divisible by 8? (a) 6929 (b) 6920 (c) 6925 (d) 6913
40. 40.What is the units digit of 7^35? (a) 1 (b) 9 (c) 7 (d) 3

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