Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 789 · 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 250 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
2. 2.What is the remainder when 478 is divided by 9? (a) 8 (b) 0 (c) 1 (d) 2
3. 3.Which of these numbers is divisible by 3? (a) 1496 (b) 1493 (c) 1494 (d) 1499
4. 4.Which of these numbers is divisible by 4? (a) 286 (b) 288 (c) 290 (d) 295
5. 5.What is the smallest number you must add to 576 to make it divisible by 10? (a) 6 (b) 5 (c) 3 (d) 4
6. 6.What is the smallest number you must add to 162 to make it divisible by 4? (a) 3 (b) 1 (c) 2
7. 7.What is the smallest number you must add to 494 to make it divisible by 7? (a) 3 (b) 4 (c) 2 (d) 5
8. 8.What is the remainder when 218 is divided by 6? (a) 2 (b) 4 (c) 1 (d) 3
9. 9.Which of these numbers is divisible by 4? (a) 1016 (b) 1015 (c) 1019 (d) 1023
10. 10.What is the units digit of 2^40? (a) 8 (b) 4 (c) 2 (d) 6
11. 11.Which of these numbers is divisible by 11? (a) 643 (b) 638 (c) 629 (d) 652
12. 12.What is the units digit of 3^39? (a) 1 (b) 3 (c) 9 (d) 7
13. 13.What is the units digit of 2^13? (a) 6 (b) 4 (c) 2 (d) 8
14. 14.Which of these numbers is divisible by 8? (a) 3663 (b) 3675 (c) 3664 (d) 3668
15. 15.What is the smallest number you must add to 173 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the remainder when 350 is divided by 8? (a) 7 (b) 6 (c) 5 (d) 2
17. 17.What is the units digit of 4^55? (a) 6 (b) 3 (c) 5 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 2474 (b) 2476 (c) 2475 (d) 2479
19. 19.What is the remainder when 453 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
20. 20.Which of these numbers is divisible by 6? (a) 4855 (b) 4860 (c) 4864 (d) 4870
21. 21.What is the remainder when 411 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
22. 22.What is the units digit of 4^51? (a) 6 (b) 5 (c) 3 (d) 4
23. 23.Which of these numbers is divisible by 8? (a) 6508 (b) 6504 (c) 6516 (d) 6501
24. 24.What is the units digit of 7^19? (a) 3 (b) 7 (c) 9 (d) 1
25. 25.What is the units digit of 2^43? (a) 2 (b) 8 (c) 6 (d) 4
26. 26.What is the units digit of 9^11? (a) 9 (b) 7 (c) 1 (d) 8
27. 27.Which of these numbers is divisible by 6? (a) 1205 (b) 1190 (c) 1198 (d) 1194
28. 28.What is the units digit of 2^15? (a) 4 (b) 2 (c) 6 (d) 8
29. 29.Which of these numbers is divisible by 6? (a) 3942 (b) 3951 (c) 3944 (d) 3941
30. 30.Which of these numbers is divisible by 4? (a) 2385 (b) 2381 (c) 2391 (d) 2384
31. 31.What is the units digit of 4^21? (a) 3 (b) 5 (c) 4 (d) 6
32. 32.What is the smallest number you must add to 118 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
33. 33.What is the smallest number you must add to 226 to make it divisible by 4? (a) 3 (b) 1 (c) 2
34. 34.What is the remainder when 152 is divided by 3? (a) 1 (b) 2 (c) 0
35. 35.Which of these numbers is divisible by 6? (a) 3302 (b) 3306 (c) 3307 (d) 3314
36. 36.What is the remainder when 205 is divided by 3? (a) 2 (b) 0 (c) 1
37. 37.Which of these numbers is divisible by 8? (a) 1464 (b) 1466 (c) 1474 (d) 1459
38. 38.What is the smallest number you must add to 229 to make it divisible by 3? (a) 2 (b) 1
39. 39.What is the units digit of 8^42? (a) 4 (b) 6 (c) 2 (d) 8
40. 40.Which of these numbers is divisible by 4? (a) 596 (b) 597 (c) 601 (d) 594

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