Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 604 · 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 232 is divided by 12? (a) 3 (b) 5 (c) 4 (d) 8
2. 2.What is the remainder when 102 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
3. 3.What is the smallest number you must add to 103 to make it divisible by 10? (a) 8 (b) 3 (c) 6 (d) 7
4. 4.What is the smallest number you must add to 321 to make it divisible by 4? (a) 1 (b) 3 (c) 2
5. 5.Which of these numbers is divisible by 9? (a) 4914 (b) 4924 (c) 4922 (d) 4911
6. 6.What is the remainder when 64 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
7. 7.What is the remainder when 149 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 4
8. 8.What is the remainder when 496 is divided by 7? (a) 6 (b) 4 (c) 1 (d) 5
9. 9.Which of these numbers is divisible by 4? (a) 814 (b) 808 (c) 809 (d) 806
10. 10.What is the remainder when 606 is divided by 11? (a) 2 (b) 0 (c) 10 (d) 1
11. 11.Which of these numbers is divisible by 8? (a) 3443 (b) 3435 (c) 3428 (d) 3432
12. 12.Which of these numbers is divisible by 4? (a) 1281 (b) 1284 (c) 1290 (d) 1287
13. 13.What is the remainder when 74 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
14. 14.Which of these numbers is divisible by 3? (a) 2416 (b) 2419 (c) 2415 (d) 2413
15. 15.Which of these numbers is divisible by 6? (a) 1729 (b) 1734 (c) 1742 (d) 1736
16. 16.What is the remainder when 81 is divided by 5? (a) 1 (b) 0 (c) 4 (d) 2
17. 17.What is the remainder when 242 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
18. 18.What is the units digit of 8^43? (a) 2 (b) 8 (c) 4 (d) 6
19. 19.What is the units digit of 2^24? (a) 4 (b) 2 (c) 8 (d) 6
20. 20.What is the units digit of 8^11? (a) 2 (b) 8 (c) 6 (d) 4
21. 21.What is the smallest number you must add to 224 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
22. 22.Which of these numbers is divisible by 9? (a) 6876 (b) 6869 (c) 6892 (d) 6879
23. 23.What is the smallest number you must add to 343 to make it divisible by 6? (a) 5 (b) 4 (c) 1 (d) 3
24. 24.What is the remainder when 92 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
25. 25.What is the remainder when 391 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
26. 26.What is the remainder when 421 is divided by 6? (a) 1 (b) 2 (c) 0 (d) 5
27. 27.What is the remainder when 192 is divided by 10? (a) 2 (b) 8 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 3? (a) 1053 (b) 1057 (c) 1052 (d) 1054
29. 29.What is the units digit of 4^33? (a) 4 (b) 6 (c) 3 (d) 5
30. 30.Which of these numbers is divisible by 3? (a) 1926 (b) 1925 (c) 1930 (d) 1927
31. 31.What is the smallest number you must add to 94 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 3
32. 32.What is the smallest number you must add to 140 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
33. 33.Which of these numbers is divisible by 9? (a) 1728 (b) 1726 (c) 1729 (d) 1742
34. 34.What is the remainder when 188 is divided by 9? (a) 1 (b) 6 (c) 7 (d) 8
35. 35.What is the units digit of 8^19? (a) 6 (b) 8 (c) 2 (d) 4
36. 36.Which of these numbers is divisible by 11? (a) 7502 (b) 7492 (c) 7522 (d) 7505
37. 37.Which of these numbers is divisible by 8? (a) 412 (b) 408 (c) 418 (d) 407
38. 38.What is the smallest number you must add to 430 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 8? (a) 1437 (b) 1424 (c) 1421 (d) 1430
40. 40.What is the units digit of 3^60? (a) 7 (b) 9 (c) 1 (d) 3

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