Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 897 · 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 6? (a) 1189 (b) 1185 (c) 1195 (d) 1188
2. 2.Which of these numbers is divisible by 11? (a) 8924 (b) 8914 (c) 8935 (d) 8921
3. 3.Which of these numbers is divisible by 6? (a) 4540 (b) 4533 (c) 4536 (d) 4547
4. 4.What is the remainder when 296 is divided by 12? (a) 4 (b) 9 (c) 7 (d) 8
5. 5.What is the remainder when 139 is divided by 3? (a) 2 (b) 0 (c) 1
6. 6.Which of these numbers is divisible by 11? (a) 1191 (b) 1188 (c) 1186 (d) 1202
7. 7.What is the remainder when 161 is divided by 3? (a) 1 (b) 2 (c) 0
8. 8.What is the smallest number you must add to 279 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 8
9. 9.Which of these numbers is divisible by 3? (a) 325 (b) 329 (c) 324 (d) 322
10. 10.What is the remainder when 109 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
11. 11.What is the units digit of 8^20? (a) 8 (b) 4 (c) 2 (d) 6
12. 12.What is the smallest number you must add to 164 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the units digit of 3^26? (a) 7 (b) 3 (c) 1 (d) 9
14. 14.Which of these numbers is divisible by 4? (a) 313 (b) 318 (c) 323 (d) 316
15. 15.What is the smallest number you must add to 170 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
16. 16.Which of these numbers is divisible by 6? (a) 2637 (b) 2628 (c) 2624 (d) 2631
17. 17.What is the units digit of 4^21? (a) 6 (b) 3 (c) 5 (d) 4
18. 18.Which of these numbers is divisible by 11? (a) 5991 (b) 5983 (c) 5984 (d) 6002
19. 19.What is the smallest number you must add to 298 to make it divisible by 8? (a) 5 (b) 7 (c) 2 (d) 6
20. 20.What is the remainder when 637 is divided by 8? (a) 4 (b) 6 (c) 3 (d) 5
21. 21.Which of these numbers is divisible by 6? (a) 609 (b) 616 (c) 606 (d) 602
22. 22.What is the smallest number you must add to 237 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
23. 23.Which of these numbers is divisible by 6? (a) 3444 (b) 3451 (c) 3443 (d) 3447
24. 24.What is the units digit of 7^50? (a) 3 (b) 7 (c) 9 (d) 1
25. 25.What is the remainder when 255 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
26. 26.What is the remainder when 334 is divided by 8? (a) 2 (b) 5 (c) 7 (d) 6
27. 27.What is the remainder when 99 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
28. 28.What is the units digit of 9^36? (a) 0 (b) 9 (c) 2 (d) 1
29. 29.What is the units digit of 4^31? (a) 6 (b) 3 (c) 5 (d) 4
30. 30.What is the units digit of 2^18? (a) 4 (b) 6 (c) 8 (d) 2
31. 31.What is the remainder when 714 is divided by 9? (a) 3 (b) 2 (c) 6 (d) 4
32. 32.What is the remainder when 394 is divided by 12? (a) 2 (b) 11 (c) 9 (d) 10
33. 33.What is the smallest number you must add to 361 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
34. 34.What is the remainder when 246 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
35. 35.Which of these numbers is divisible by 6? (a) 2855 (b) 2844 (c) 2847 (d) 2839
36. 36.What is the units digit of 7^25? (a) 9 (b) 3 (c) 7 (d) 1
37. 37.Which of these numbers is divisible by 11? (a) 6711 (b) 6724 (c) 6710 (d) 6709
38. 38.Which of these numbers is divisible by 3? (a) 2687 (b) 2682 (c) 2680 (d) 2684
39. 39.Which of these numbers is divisible by 4? (a) 545 (b) 551 (c) 554 (d) 548
40. 40.What is the units digit of 9^21? (a) 7 (b) 9 (c) 8 (d) 1

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