Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 295 · 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 559 to make it divisible by 12? (a) 4 (b) 7 (c) 5 (d) 6
2. 2.Which of these numbers is divisible by 6? (a) 4874 (b) 4878 (c) 4880 (d) 4885
3. 3.What is the remainder when 138 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
4. 4.What is the smallest number you must add to 81 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
5. 5.Which of these numbers is divisible by 8? (a) 4752 (b) 4766 (c) 4757 (d) 4746
6. 6.What is the smallest number you must add to 449 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 1
7. 7.What is the smallest number you must add to 129 to make it divisible by 4? (a) 1 (b) 2 (c) 3
8. 8.What is the units digit of 8^20? (a) 2 (b) 8 (c) 6 (d) 4
9. 9.What is the smallest number you must add to 139 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the smallest number you must add to 389 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
11. 11.What is the smallest number you must add to 208 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the remainder when 117 is divided by 10? (a) 8 (b) 6 (c) 3 (d) 7
13. 13.Which of these numbers is divisible by 11? (a) 6784 (b) 6787 (c) 6801 (d) 6797
14. 14.What is the remainder when 212 is divided by 3? (a) 0 (b) 1 (c) 2
15. 15.What is the units digit of 3^54? (a) 3 (b) 7 (c) 9 (d) 1
16. 16.What is the smallest number you must add to 241 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
17. 17.What is the units digit of 9^13? (a) 1 (b) 9 (c) 8 (d) 7
18. 18.What is the smallest number you must add to 115 to make it divisible by 3? (a) 1 (b) 2
19. 19.What is the units digit of 2^18? (a) 8 (b) 2 (c) 6 (d) 4
20. 20.What is the units digit of 2^42? (a) 2 (b) 8 (c) 6 (d) 4
21. 21.What is the units digit of 2^41? (a) 8 (b) 6 (c) 2 (d) 4
22. 22.Which of these numbers is divisible by 4? (a) 2721 (b) 2716 (c) 2718 (d) 2713
23. 23.What is the remainder when 187 is divided by 6? (a) 1 (b) 5 (c) 2 (d) 0
24. 24.What is the smallest number you must add to 91 to make it divisible by 3? (a) 2 (b) 1
25. 25.Which of these numbers is divisible by 4? (a) 1685 (b) 1682 (c) 1678 (d) 1680
26. 26.What is the remainder when 333 is divided by 7? (a) 6 (b) 3 (c) 4 (d) 5
27. 27.What is the units digit of 4^28? (a) 5 (b) 4 (c) 6 (d) 7
28. 28.What is the remainder when 540 is divided by 7? (a) 2 (b) 6 (c) 1 (d) 0
29. 29.Which of these numbers is divisible by 6? (a) 4236 (b) 4245 (c) 4237 (d) 4234
30. 30.What is the smallest number you must add to 159 to make it divisible by 12? (a) 3 (b) 9 (c) 10 (d) 8
31. 31.Which of these numbers is divisible by 6? (a) 1521 (b) 1528 (c) 1516 (d) 1518
32. 32.Which of these numbers is divisible by 6? (a) 2833 (b) 2838 (c) 2849 (d) 2843
33. 33.What is the smallest number you must add to 437 to make it divisible by 9? (a) 5 (b) 4 (c) 6 (d) 3
34. 34.Which of these numbers is divisible by 9? (a) 4133 (b) 4146 (c) 4140 (d) 4153
35. 35.What is the units digit of 8^13? (a) 2 (b) 4 (c) 6 (d) 8
36. 36.What is the smallest number you must add to 415 to make it divisible by 8? (a) 7 (b) 3 (c) 2 (d) 1
37. 37.What is the remainder when 165 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
38. 38.What is the smallest number you must add to 853 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 1
39. 39.Which of these numbers is divisible by 4? (a) 558 (b) 550 (c) 552 (d) 555
40. 40.Which of these numbers is divisible by 11? (a) 6478 (b) 6462 (c) 6481 (d) 6468

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