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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 865 · 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 691 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
  2. 2.What is the remainder when 907 is divided by 12? (a) 6 (b) 5 (c) 8 (d) 7
  3. 3.What is the smallest number you must add to 228 to make it divisible by 11? (a) 2 (b) 4 (c) 3 (d) 8
  4. 4.Which of these numbers is divisible by 8? (a) 3633 (b) 3641 (c) 3640 (d) 3654
  5. 5.What is the remainder when 86 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
  6. 6.What is the remainder when 132 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
  7. 7.What is the units digit of 3^18? (a) 1 (b) 7 (c) 9 (d) 3
  8. 8.What is the remainder when 817 is divided by 11? (a) 8 (b) 2 (c) 3 (d) 4
  9. 9.What is the smallest number you must add to 31 to make it divisible by 3? (a) 2 (b) 1
  10. 10.What is the smallest number you must add to 217 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  11. 11.Which of these numbers is divisible by 4? (a) 577 (b) 571 (c) 572 (d) 575
  12. 12.What is the units digit of 8^59? (a) 2 (b) 6 (c) 8 (d) 4
  13. 13.What is the smallest number you must add to 137 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
  14. 14.What is the smallest number you must add to 111 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  15. 15.What is the remainder when 223 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
  16. 16.Which of these numbers is divisible by 3? (a) 657 (b) 662 (c) 656 (d) 659
  17. 17.Which of these numbers is divisible by 9? (a) 5768 (b) 5751 (c) 5746 (d) 5756
  18. 18.What is the smallest number you must add to 875 to make it divisible by 11? (a) 7 (b) 5 (c) 4 (d) 6
  19. 19.What is the remainder when 158 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 3
  20. 20.What is the remainder when 135 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  21. 21.What is the smallest number you must add to 235 to make it divisible by 3? (a) 1 (b) 2
  22. 22.Which of these numbers is divisible by 11? (a) 6155 (b) 6144 (c) 6162 (d) 6149
  23. 23.What is the smallest number you must add to 127 to make it divisible by 7? (a) 1 (b) 5 (c) 6 (d) 4
  24. 24.What is the remainder when 157 is divided by 3? (a) 0 (b) 2 (c) 1
  25. 25.What is the remainder when 231 is divided by 9? (a) 3 (b) 7 (c) 5 (d) 6
  26. 26.Which of these numbers is divisible by 8? (a) 2116 (b) 2104 (c) 2103 (d) 2108
  27. 27.What is the remainder when 263 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  28. 28.What is the smallest number you must add to 238 to make it divisible by 9? (a) 4 (b) 7 (c) 6 (d) 5
  29. 29.What is the smallest number you must add to 255 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
  30. 30.What is the units digit of 9^28? (a) 0 (b) 1 (c) 2 (d) 9
  31. 31.Which of these numbers is divisible by 11? (a) 539 (b) 529 (c) 540 (d) 553
  32. 32.What is the remainder when 392 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
  33. 33.What is the units digit of 8^44? (a) 2 (b) 4 (c) 8 (d) 6
  34. 34.What is the smallest number you must add to 291 to make it divisible by 8? (a) 4 (b) 6 (c) 3 (d) 5
  35. 35.What is the remainder when 581 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
  36. 36.What is the remainder when 301 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 0
  37. 37.What is the remainder when 182 is divided by 12? (a) 10 (b) 2 (c) 1 (d) 3
  38. 38.What is the remainder when 239 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
  39. 39.What is the smallest number you must add to 143 to make it divisible by 3? (a) 1 (b) 2
  40. 40.What is the smallest number you must add to 202 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3

Answer key

  1. 1
  2. 7
  3. 3
  4. 3640
  5. 2
  6. 4
  7. 9
  8. 3
  9. 2
  10. 3
  11. 572
  12. 2
  13. 7
  14. 1
  15. 3
  16. 657
  17. 5751
  18. 5
  19. 4
  20. 3
  21. 2
  22. 6149
  23. 6
  24. 1
  25. 6
  26. 2104
  27. 3
  28. 5
  29. 1
  30. 1
  31. 539
  32. 2
  33. 6
  34. 5
  35. 1
  36. 1
  37. 2
  38. 5
  39. 1
  40. 3

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