Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 932 · 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 509 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
2. 2.What is the smallest number you must add to 165 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
3. 3.What is the remainder when 154 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
4. 4.What is the units digit of 9^17? (a) 1 (b) 8 (c) 9 (d) 7
5. 5.What is the units digit of 3^26? (a) 3 (b) 1 (c) 9 (d) 7
6. 6.Which of these numbers is divisible by 6? (a) 388 (b) 378 (c) 375 (d) 382
7. 7.What is the remainder when 99 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
8. 8.Which of these numbers is divisible by 9? (a) 1249 (b) 1256 (c) 1238 (d) 1242
9. 9.What is the remainder when 689 is divided by 10? (a) 8 (b) 7 (c) 1 (d) 9
10. 10.What is the units digit of 3^25? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.What is the units digit of 7^15? (a) 3 (b) 7 (c) 9 (d) 1
12. 12.What is the remainder when 143 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
13. 13.What is the units digit of 3^14? (a) 9 (b) 7 (c) 3 (d) 1
14. 14.What is the remainder when 314 is divided by 12? (a) 10 (b) 3 (c) 2 (d) 1
15. 15.What is the remainder when 261 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
16. 16.What is the remainder when 677 is divided by 10? (a) 3 (b) 8 (c) 7 (d) 6
17. 17.What is the smallest number you must add to 601 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
18. 18.What is the smallest number you must add to 407 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 4? (a) 2827 (b) 2822 (c) 2820 (d) 2817
20. 20.Which of these numbers is divisible by 9? (a) 4456 (b) 4455 (c) 4448 (d) 4469
21. 21.Which of these numbers is divisible by 9? (a) 7164 (b) 7163 (c) 7168 (d) 7176
22. 22.Which of these numbers is divisible by 3? (a) 358 (b) 356 (c) 354 (d) 353
23. 23.What is the units digit of 4^11? (a) 5 (b) 4 (c) 3 (d) 6
24. 24.What is the units digit of 9^26? (a) 9 (b) 2 (c) 0 (d) 1
25. 25.Which of these numbers is divisible by 4? (a) 874 (b) 869 (c) 868 (d) 866
26. 26.What is the remainder when 173 is divided by 11? (a) 7 (b) 9 (c) 8 (d) 3
27. 27.What is the remainder when 805 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
28. 28.Which of these numbers is divisible by 9? (a) 1327 (b) 1320 (c) 1340 (d) 1323
29. 29.Which of these numbers is divisible by 3? (a) 521 (b) 519 (c) 518 (d) 523
30. 30.What is the smallest number you must add to 125 to make it divisible by 9? (a) 2 (b) 8 (c) 1 (d) 3
31. 31.What is the units digit of 9^51? (a) 8 (b) 7 (c) 1 (d) 9
32. 32.What is the remainder when 103 is divided by 8? (a) 7 (b) 1 (c) 6 (d) 5
33. 33.What is the units digit of 7^37? (a) 7 (b) 9 (c) 1 (d) 3
34. 34.What is the units digit of 3^12? (a) 3 (b) 1 (c) 7 (d) 9
35. 35.Which of these numbers is divisible by 6? (a) 5244 (b) 5255 (c) 5241 (d) 5245
36. 36.Which of these numbers is divisible by 6? (a) 1426 (b) 1428 (c) 1429 (d) 1438
37. 37.Which of these numbers is divisible by 8? (a) 3520 (b) 3530 (c) 3522 (d) 3517
38. 38.What is the smallest number you must add to 459 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
39. 39.What is the remainder when 628 is divided by 9? (a) 7 (b) 6 (c) 2 (d) 8
40. 40.Which of these numbers is divisible by 9? (a) 5787 (b) 5786 (c) 5803 (d) 5795

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