Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 593 · 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 smallest number you must add to 49 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the smallest number you must add to 324 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
3. 3.What is the remainder when 498 is divided by 8? (a) 3 (b) 1 (c) 2 (d) 6
4. 4.What is the remainder when 91 is divided by 6? (a) 0 (b) 5 (c) 1 (d) 2
5. 5.Which of these numbers is divisible by 3? (a) 769 (b) 768 (c) 773 (d) 766
6. 6.Which of these numbers is divisible by 6? (a) 4698 (b) 4709 (c) 4699 (d) 4696
7. 7.Which of these numbers is divisible by 8? (a) 6982 (b) 6984 (c) 6991 (d) 6993
8. 8.What is the smallest number you must add to 335 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
9. 9.Which of these numbers is divisible by 8? (a) 1728 (b) 1741 (c) 1735 (d) 1726
10. 10.What is the units digit of 4^43? (a) 6 (b) 5 (c) 4 (d) 3
11. 11.What is the smallest number you must add to 38 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the smallest number you must add to 200 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the remainder when 661 is divided by 9? (a) 4 (b) 5 (c) 3 (d) 6
14. 14.Which of these numbers is divisible by 8? (a) 2080 (b) 2094 (c) 2081 (d) 2073
15. 15.What is the units digit of 9^32? (a) 1 (b) 2 (c) 0 (d) 9
16. 16.Which of these numbers is divisible by 4? (a) 1367 (b) 1364 (c) 1369 (d) 1363
17. 17.What is the remainder when 565 is divided by 8? (a) 3 (b) 4 (c) 5 (d) 6
18. 18.Which of these numbers is divisible by 11? (a) 5744 (b) 5742 (c) 5756 (d) 5737
19. 19.Which of these numbers is divisible by 9? (a) 518 (b) 528 (c) 534 (d) 522
20. 20.Which of these numbers is divisible by 8? (a) 5698 (b) 5717 (c) 5706 (d) 5704
21. 21.Which of these numbers is divisible by 9? (a) 2349 (b) 2348 (c) 2365 (d) 2351
22. 22.What is the units digit of 2^12? (a) 4 (b) 2 (c) 6 (d) 8
23. 23.Which of these numbers is divisible by 4? (a) 725 (b) 730 (c) 723 (d) 724
24. 24.Which of these numbers is divisible by 6? (a) 4706 (b) 4710 (c) 4714 (d) 4717
25. 25.What is the smallest number you must add to 483 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
26. 26.Which of these numbers is divisible by 3? (a) 181 (b) 184 (c) 178 (d) 180
27. 27.Which of these numbers is divisible by 9? (a) 2934 (b) 2945 (c) 2933 (d) 2937
28. 28.What is the remainder when 300 is divided by 7? (a) 5 (b) 4 (c) 1 (d) 6
29. 29.What is the units digit of 4^15? (a) 6 (b) 5 (c) 4 (d) 3
30. 30.What is the smallest number you must add to 163 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
31. 31.Which of these numbers is divisible by 8? (a) 1791 (b) 1792 (c) 1798 (d) 1801
32. 32.What is the remainder when 86 is divided by 3? (a) 1 (b) 0 (c) 2
33. 33.Which of these numbers is divisible by 3? (a) 514 (b) 513 (c) 517 (d) 512
34. 34.What is the smallest number you must add to 516 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
35. 35.What is the remainder when 233 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
36. 36.What is the units digit of 2^11? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.What is the smallest number you must add to 519 to make it divisible by 12? (a) 10 (b) 3 (c) 9 (d) 8
38. 38.Which of these numbers is divisible by 11? (a) 2659 (b) 2683 (c) 2669 (d) 2662
39. 39.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
40. 40.What is the smallest number you must add to 287 to make it divisible by 4? (a) 3 (b) 1 (c) 2

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