Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 250 · 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 units digit of 3^38? (a) 7 (b) 9 (c) 1 (d) 3
2. 2.What is the smallest number you must add to 307 to make it divisible by 4? (a) 2 (b) 1 (c) 3
3. 3.What is the remainder when 360 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 4
4. 4.What is the remainder when 356 is divided by 7? (a) 5 (b) 4 (c) 1 (d) 6
5. 5.What is the units digit of 2^39? (a) 6 (b) 2 (c) 8 (d) 4
6. 6.What is the remainder when 661 is divided by 10? (a) 1 (b) 9 (c) 2 (d) 0
7. 7.Which of these numbers is divisible by 3? (a) 1536 (b) 1535 (c) 1540 (d) 1538
8. 8.What is the smallest number you must add to 291 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
9. 9.Which of these numbers is divisible by 4? (a) 1022 (b) 1024 (c) 1027 (d) 1029
10. 10.What is the units digit of 2^16? (a) 2 (b) 4 (c) 8 (d) 6
11. 11.What is the remainder when 58 is divided by 3? (a) 0 (b) 2 (c) 1
12. 12.What is the units digit of 7^15? (a) 3 (b) 1 (c) 9 (d) 7
13. 13.What is the smallest number you must add to 446 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 5
14. 14.What is the units digit of 7^14? (a) 3 (b) 7 (c) 1 (d) 9
15. 15.What is the smallest number you must add to 149 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
16. 16.Which of these numbers is divisible by 6? (a) 3110 (b) 3106 (c) 3117 (d) 3108
17. 17.What is the units digit of 8^19? (a) 8 (b) 2 (c) 6 (d) 4
18. 18.Which of these numbers is divisible by 6? (a) 1801 (b) 1796 (c) 1800 (d) 1809
19. 19.What is the remainder when 363 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
20. 20.What is the units digit of 2^30? (a) 8 (b) 4 (c) 2 (d) 6
21. 21.What is the smallest number you must add to 677 to make it divisible by 12? (a) 5 (b) 7 (c) 8 (d) 6
22. 22.What is the smallest number you must add to 337 to make it divisible by 11? (a) 5 (b) 7 (c) 3 (d) 4
23. 23.What is the smallest number you must add to 103 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
24. 24.What is the smallest number you must add to 323 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 8
25. 25.What is the remainder when 331 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
26. 26.Which of these numbers is divisible by 9? (a) 823 (b) 835 (c) 828 (d) 844
27. 27.What is the remainder when 126 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
28. 28.Which of these numbers is divisible by 6? (a) 3572 (b) 3560 (c) 3567 (d) 3564
29. 29.Which of these numbers is divisible by 3? (a) 1000 (b) 994 (c) 996 (d) 997
30. 30.What is the smallest number you must add to 547 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 1
31. 31.What is the units digit of 4^25? (a) 4 (b) 5 (c) 6 (d) 3
32. 32.What is the smallest number you must add to 211 to make it divisible by 3? (a) 2 (b) 1
33. 33.What is the remainder when 510 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 1
34. 34.What is the units digit of 3^24? (a) 1 (b) 3 (c) 9 (d) 7
35. 35.Which of these numbers is divisible by 11? (a) 9828 (b) 9821 (c) 9806 (d) 9812
36. 36.What is the remainder when 399 is divided by 9? (a) 4 (b) 2 (c) 3 (d) 6
37. 37.What is the remainder when 218 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
38. 38.What is the remainder when 347 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
39. 39.What is the remainder when 128 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
40. 40.What is the remainder when 74 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4

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