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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 630 · 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.What is the remainder when 325 is divided by 12? (a) 0 (b) 2 (c) 11 (d) 1
  2. 2.Which of these numbers is divisible by 8? (a) 6964 (b) 6969 (c) 6959 (d) 6960
  3. 3.What is the smallest number you must add to 270 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 4
  4. 4.What is the units digit of 2^49? (a) 6 (b) 8 (c) 4 (d) 2
  5. 5.What is the smallest number you must add to 293 to make it divisible by 11? (a) 3 (b) 5 (c) 7 (d) 4
  6. 6.What is the smallest number you must add to 381 to make it divisible by 12? (a) 9 (b) 2 (c) 3 (d) 4
  7. 7.What is the units digit of 3^18? (a) 3 (b) 1 (c) 7 (d) 9
  8. 8.Which of these numbers is divisible by 6? (a) 809 (b) 811 (c) 810 (d) 817
  9. 9.What is the smallest number you must add to 298 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 2
  10. 10.What is the smallest number you must add to 195 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  11. 11.What is the smallest number you must add to 776 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
  12. 12.What is the units digit of 3^19? (a) 3 (b) 1 (c) 9 (d) 7
  13. 13.What is the smallest number you must add to 359 to make it divisible by 9? (a) 8 (b) 2 (c) 3 (d) 1
  14. 14.What is the units digit of 7^12? (a) 3 (b) 7 (c) 9 (d) 1
  15. 15.What is the units digit of 2^16? (a) 2 (b) 8 (c) 4 (d) 6
  16. 16.What is the remainder when 219 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
  17. 17.What is the smallest number you must add to 40 to make it divisible by 3? (a) 2 (b) 1
  18. 18.What is the units digit of 3^54? (a) 7 (b) 1 (c) 9 (d) 3
  19. 19.What is the remainder when 291 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
  20. 20.Which of these numbers is divisible by 4? (a) 2421 (b) 2424 (c) 2427 (d) 2430
  21. 21.What is the remainder when 145 is divided by 11? (a) 9 (b) 3 (c) 1 (d) 2
  22. 22.What is the units digit of 8^20? (a) 2 (b) 6 (c) 4 (d) 8
  23. 23.What is the remainder when 486 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
  24. 24.Which of these numbers is divisible by 11? (a) 1884 (b) 1881 (c) 1879 (d) 1899
  25. 25.Which of these numbers is divisible by 9? (a) 733 (b) 722 (c) 720 (d) 719
  26. 26.What is the smallest number you must add to 508 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
  27. 27.What is the units digit of 7^20? (a) 9 (b) 7 (c) 1 (d) 3
  28. 28.Which of these numbers is divisible by 3? (a) 1726 (b) 1730 (c) 1728 (d) 1733
  29. 29.Which of these numbers is divisible by 6? (a) 1063 (b) 1069 (c) 1076 (d) 1068
  30. 30.What is the smallest number you must add to 193 to make it divisible by 12? (a) 11 (b) 9 (c) 1 (d) 10
  31. 31.What is the smallest number you must add to 392 to make it divisible by 11? (a) 7 (b) 5 (c) 3 (d) 4
  32. 32.What is the remainder when 721 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
  33. 33.Which of these numbers is divisible by 9? (a) 5634 (b) 5632 (c) 5651 (d) 5641
  34. 34.Which of these numbers is divisible by 6? (a) 2478 (b) 2477 (c) 2481 (d) 2487
  35. 35.What is the smallest number you must add to 837 to make it divisible by 11? (a) 10 (b) 1 (c) 9 (d) 8
  36. 36.What is the smallest number you must add to 277 to make it divisible by 10? (a) 4 (b) 7 (c) 3 (d) 2
  37. 37.Which of these numbers is divisible by 6? (a) 2122 (b) 2108 (c) 2112 (d) 2116
  38. 38.What is the smallest number you must add to 485 to make it divisible by 7? (a) 2 (b) 4 (c) 6 (d) 5
  39. 39.What is the remainder when 785 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
  40. 40.What is the units digit of 3^21? (a) 1 (b) 9 (c) 7 (d) 3

Answer key

  1. 1
  2. 6960
  3. 5
  4. 2
  5. 4
  6. 3
  7. 9
  8. 810
  9. 2
  10. 1
  11. 4
  12. 7
  13. 1
  14. 1
  15. 6
  16. 4
  17. 2
  18. 9
  19. 1
  20. 2424
  21. 2
  22. 6
  23. 3
  24. 1881
  25. 720
  26. 4
  27. 1
  28. 1728
  29. 1068
  30. 11
  31. 4
  32. 1
  33. 5634
  34. 2478
  35. 10
  36. 3
  37. 2112
  38. 5
  39. 5
  40. 3

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