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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 946 · 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 322 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
  2. 2.What is the remainder when 155 is divided by 3? (a) 1 (b) 0 (c) 2
  3. 3.What is the smallest number you must add to 212 to make it divisible by 3? (a) 1 (b) 2
  4. 4.What is the smallest number you must add to 398 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 2
  5. 5.What is the smallest number you must add to 637 to make it divisible by 11? (a) 2 (b) 1 (c) 3 (d) 10
  6. 6.What is the smallest number you must add to 293 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  7. 7.What is the smallest number you must add to 82 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  8. 8.What is the remainder when 757 is divided by 12? (a) 2 (b) 1 (c) 0 (d) 11
  9. 9.What is the smallest number you must add to 88 to make it divisible by 3? (a) 2 (b) 1
  10. 10.What is the smallest number you must add to 877 to make it divisible by 11? (a) 8 (b) 4 (c) 3 (d) 2
  11. 11.What is the remainder when 529 is divided by 7? (a) 3 (b) 5 (c) 6 (d) 4
  12. 12.Which of these numbers is divisible by 9? (a) 1694 (b) 1683 (c) 1681 (d) 1685
  13. 13.Which of these numbers is divisible by 6? (a) 886 (b) 888 (c) 895 (d) 892
  14. 14.What is the remainder when 533 is divided by 12? (a) 4 (b) 5 (c) 6 (d) 7
  15. 15.What is the remainder when 118 is divided by 3? (a) 2 (b) 1 (c) 0
  16. 16.What is the remainder when 328 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 5
  17. 17.What is the smallest number you must add to 220 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 4
  18. 18.What is the smallest number you must add to 344 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
  19. 19.What is the smallest number you must add to 561 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
  20. 20.What is the units digit of 3^46? (a) 1 (b) 3 (c) 9 (d) 7
  21. 21.What is the smallest number you must add to 689 to make it divisible by 11? (a) 3 (b) 4 (c) 7 (d) 5
  22. 22.What is the smallest number you must add to 131 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  23. 23.Which of these numbers is divisible by 8? (a) 6426 (b) 6432 (c) 6439 (d) 6444
  24. 24.What is the units digit of 3^49? (a) 7 (b) 1 (c) 9 (d) 3
  25. 25.What is the smallest number you must add to 182 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
  26. 26.What is the remainder when 114 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 7
  27. 27.What is the smallest number you must add to 258 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  28. 28.What is the smallest number you must add to 78 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  29. 29.What is the smallest number you must add to 138 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  30. 30.What is the units digit of 3^24? (a) 3 (b) 7 (c) 9 (d) 1
  31. 31.Which of these numbers is divisible by 4? (a) 3006 (b) 2997 (c) 3000 (d) 3003
  32. 32.What is the units digit of 2^22? (a) 4 (b) 8 (c) 6 (d) 2
  33. 33.Which of these numbers is divisible by 3? (a) 221 (b) 215 (c) 216 (d) 217
  34. 34.What is the smallest number you must add to 181 to make it divisible by 12? (a) 1 (b) 9 (c) 11 (d) 10
  35. 35.Which of these numbers is divisible by 8? (a) 902 (b) 891 (c) 888 (d) 887
  36. 36.Which of these numbers is divisible by 8? (a) 2629 (b) 2637 (c) 2644 (d) 2632
  37. 37.What is the smallest number you must add to 143 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  38. 38.What is the units digit of 4^47? (a) 4 (b) 6 (c) 5 (d) 3
  39. 39.What is the smallest number you must add to 475 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
  40. 40.What is the smallest number you must add to 243 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5

Answer key

  1. 2
  2. 2
  3. 1
  4. 7
  5. 1
  6. 3
  7. 3
  8. 1
  9. 2
  10. 3
  11. 4
  12. 1683
  13. 888
  14. 5
  15. 1
  16. 4
  17. 2
  18. 1
  19. 6
  20. 9
  21. 4
  22. 4
  23. 6432
  24. 3
  25. 4
  26. 6
  27. 2
  28. 2
  29. 2
  30. 1
  31. 3000
  32. 4
  33. 216
  34. 11
  35. 888
  36. 2632
  37. 1
  38. 4
  39. 5
  40. 5

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