← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 394 · 40 questions
TalentJR

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.

NameDateTime takenScore ___ / 40
  1. 1.Which of these numbers is divisible by 4? (a) 1664 (b) 1662 (c) 1665 (d) 1671
  2. 2.Which of these numbers is divisible by 3? (a) 460 (b) 457 (c) 455 (d) 456
  3. 3.What is the remainder when 182 is divided by 3? (a) 2 (b) 1 (c) 0
  4. 4.What is the remainder when 509 is divided by 12? (a) 7 (b) 5 (c) 6 (d) 4
  5. 5.What is the remainder when 517 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
  6. 6.What is the remainder when 616 is divided by 9? (a) 6 (b) 5 (c) 4 (d) 3
  7. 7.Which of these numbers is divisible by 9? (a) 1798 (b) 1789 (c) 1775 (d) 1782
  8. 8.What is the smallest number you must add to 606 to make it divisible by 11? (a) 10 (b) 1 (c) 8 (d) 9
  9. 9.What is the smallest number you must add to 128 to make it divisible by 10? (a) 3 (b) 1 (c) 8 (d) 2
  10. 10.What is the smallest number you must add to 59 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  11. 11.Which of these numbers is divisible by 11? (a) 3982 (b) 3991 (c) 4002 (d) 3978
  12. 12.What is the smallest number you must add to 125 to make it divisible by 8? (a) 2 (b) 4 (c) 5 (d) 3
  13. 13.Which of these numbers is divisible by 4? (a) 721 (b) 719 (c) 720 (d) 725
  14. 14.What is the smallest number you must add to 473 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 2
  15. 15.What is the remainder when 567 is divided by 8? (a) 1 (b) 7 (c) 6 (d) 5
  16. 16.Which of these numbers is divisible by 4? (a) 690 (b) 698 (c) 695 (d) 692
  17. 17.What is the remainder when 621 is divided by 10? (a) 1 (b) 0 (c) 9 (d) 2
  18. 18.Which of these numbers is divisible by 9? (a) 6255 (b) 6256 (c) 6272 (d) 6251
  19. 19.Which of these numbers is divisible by 4? (a) 775 (b) 779 (c) 776 (d) 782
  20. 20.What is the smallest number you must add to 333 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 2
  21. 21.What is the remainder when 413 is divided by 6? (a) 5 (b) 3 (c) 1 (d) 4
  22. 22.What is the smallest number you must add to 194 to make it divisible by 10? (a) 6 (b) 4 (c) 5 (d) 7
  23. 23.What is the remainder when 484 is divided by 7? (a) 0 (b) 2 (c) 1 (d) 6
  24. 24.What is the smallest number you must add to 231 to make it divisible by 12? (a) 3 (b) 10 (c) 8 (d) 9
  25. 25.What is the units digit of 8^55? (a) 8 (b) 4 (c) 2 (d) 6
  26. 26.What is the units digit of 3^39? (a) 3 (b) 9 (c) 7 (d) 1
  27. 27.Which of these numbers is divisible by 4? (a) 2636 (b) 2643 (c) 2634 (d) 2637
  28. 28.What is the remainder when 488 is divided by 9? (a) 2 (b) 3 (c) 7 (d) 1
  29. 29.What is the remainder when 575 is divided by 11? (a) 4 (b) 8 (c) 3 (d) 2
  30. 30.What is the smallest number you must add to 720 to make it divisible by 11? (a) 7 (b) 8 (c) 6 (d) 5
  31. 31.What is the smallest number you must add to 677 to make it divisible by 9? (a) 6 (b) 2 (c) 7 (d) 8
  32. 32.What is the units digit of 4^35? (a) 3 (b) 4 (c) 6 (d) 5
  33. 33.What is the units digit of 9^60? (a) 1 (b) 9 (c) 0 (d) 2
  34. 34.What is the remainder when 374 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 4
  35. 35.What is the remainder when 307 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
  36. 36.What is the remainder when 722 is divided by 9? (a) 3 (b) 1 (c) 2 (d) 7
  37. 37.What is the smallest number you must add to 213 to make it divisible by 12? (a) 4 (b) 9 (c) 3 (d) 2
  38. 38.Which of these numbers is divisible by 3? (a) 1634 (b) 1630 (c) 1629 (d) 1627
  39. 39.What is the smallest number you must add to 213 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  40. 40.What is the remainder when 477 is divided by 10? (a) 7 (b) 6 (c) 8 (d) 3

Answer key

  1. 1664
  2. 456
  3. 2
  4. 5
  5. 6
  6. 4
  7. 1782
  8. 10
  9. 2
  10. 1
  11. 3982
  12. 3
  13. 720
  14. 1
  15. 7
  16. 692
  17. 1
  18. 6255
  19. 776
  20. 3
  21. 5
  22. 6
  23. 1
  24. 9
  25. 2
  26. 7
  27. 2636
  28. 2
  29. 3
  30. 6
  31. 7
  32. 4
  33. 1
  34. 5
  35. 3
  36. 2
  37. 3
  38. 1629
  39. 3
  40. 7

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