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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 473 · 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 6? (a) 1794 (b) 1797 (c) 1791 (d) 1805
  2. 2.What is the smallest number you must add to 783 to make it divisible by 11? (a) 8 (b) 9 (c) 2 (d) 10
  3. 3.What is the units digit of 4^49? (a) 4 (b) 6 (c) 5 (d) 3
  4. 4.What is the remainder when 388 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
  5. 5.What is the smallest number you must add to 873 to make it divisible by 12? (a) 2 (b) 3 (c) 4 (d) 9
  6. 6.Which of these numbers is divisible by 3? (a) 1869 (b) 1873 (c) 1868 (d) 1871
  7. 7.What is the smallest number you must add to 113 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  8. 8.What is the units digit of 9^25? (a) 7 (b) 1 (c) 8 (d) 9
  9. 9.Which of these numbers is divisible by 6? (a) 2772 (b) 2773 (c) 2767 (d) 2780
  10. 10.What is the remainder when 512 is divided by 7? (a) 2 (b) 1 (c) 0 (d) 6
  11. 11.Which of these numbers is divisible by 11? (a) 3619 (b) 3638 (c) 3629 (d) 3615
  12. 12.What is the remainder when 285 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
  13. 13.What is the units digit of 7^18? (a) 1 (b) 3 (c) 9 (d) 7
  14. 14.What is the remainder when 387 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 2
  15. 15.What is the remainder when 238 is divided by 3? (a) 1 (b) 0 (c) 2
  16. 16.What is the smallest number you must add to 229 to make it divisible by 3? (a) 1 (b) 2
  17. 17.What is the smallest number you must add to 192 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  18. 18.What is the remainder when 230 is divided by 11? (a) 1 (b) 9 (c) 8 (d) 10
  19. 19.What is the units digit of 7^40? (a) 3 (b) 7 (c) 1 (d) 9
  20. 20.Which of these numbers is divisible by 4? (a) 1893 (b) 1892 (c) 1899 (d) 1890
  21. 21.What is the units digit of 9^28? (a) 0 (b) 9 (c) 1 (d) 2
  22. 22.What is the remainder when 268 is divided by 9? (a) 2 (b) 6 (c) 7 (d) 8
  23. 23.What is the smallest number you must add to 359 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 1
  24. 24.Which of these numbers is divisible by 11? (a) 4037 (b) 4032 (c) 4055 (d) 4046
  25. 25.What is the units digit of 8^45? (a) 6 (b) 2 (c) 4 (d) 8
  26. 26.What is the units digit of 9^31? (a) 8 (b) 1 (c) 9 (d) 7
  27. 27.What is the smallest number you must add to 453 to make it divisible by 10? (a) 3 (b) 6 (c) 7 (d) 8
  28. 28.What is the remainder when 317 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
  29. 29.What is the smallest number you must add to 445 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
  30. 30.What is the units digit of 9^18? (a) 9 (b) 1 (c) 2 (d) 0
  31. 31.What is the units digit of 9^36? (a) 1 (b) 9 (c) 2 (d) 0
  32. 32.What is the units digit of 4^25? (a) 6 (b) 4 (c) 3 (d) 5
  33. 33.Which of these numbers is divisible by 9? (a) 5984 (b) 5976 (c) 5975 (d) 5986
  34. 34.What is the smallest number you must add to 156 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  35. 35.What is the units digit of 8^12? (a) 8 (b) 6 (c) 2 (d) 4
  36. 36.What is the remainder when 139 is divided by 3? (a) 2 (b) 0 (c) 1
  37. 37.What is the remainder when 250 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  38. 38.What is the smallest number you must add to 421 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
  39. 39.What is the units digit of 3^20? (a) 3 (b) 9 (c) 1 (d) 7
  40. 40.What is the remainder when 362 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3

Answer key

  1. 1794
  2. 9
  3. 4
  4. 3
  5. 3
  6. 1869
  7. 3
  8. 9
  9. 2772
  10. 1
  11. 3619
  12. 1
  13. 9
  14. 3
  15. 1
  16. 2
  17. 3
  18. 10
  19. 1
  20. 1892
  21. 1
  22. 7
  23. 1
  24. 4037
  25. 8
  26. 9
  27. 7
  28. 1
  29. 5
  30. 1
  31. 1
  32. 4
  33. 5976
  34. 4
  35. 6
  36. 1
  37. 2
  38. 5
  39. 1
  40. 2

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