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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 510 · 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 3? (a) 823 (b) 822 (c) 827 (d) 820
  2. 2.What is the remainder when 106 is divided by 3? (a) 0 (b) 1 (c) 2
  3. 3.What is the smallest number you must add to 225 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  4. 4.What is the remainder when 172 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
  5. 5.Which of these numbers is divisible by 9? (a) 6042 (b) 6036 (c) 6025 (d) 6030
  6. 6.Which of these numbers is divisible by 3? (a) 1787 (b) 1784 (c) 1781 (d) 1782
  7. 7.What is the remainder when 492 is divided by 9? (a) 5 (b) 7 (c) 3 (d) 6
  8. 8.Which of these numbers is divisible by 11? (a) 485 (b) 482 (c) 472 (d) 473
  9. 9.What is the smallest number you must add to 860 to make it divisible by 12? (a) 3 (b) 5 (c) 8 (d) 4
  10. 10.What is the remainder when 134 is divided by 8? (a) 5 (b) 6 (c) 2 (d) 7
  11. 11.What is the units digit of 3^42? (a) 3 (b) 1 (c) 7 (d) 9
  12. 12.Which of these numbers is divisible by 11? (a) 2071 (b) 2079 (c) 2081 (d) 2097
  13. 13.What is the remainder when 294 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
  14. 14.What is the remainder when 634 is divided by 11? (a) 6 (b) 4 (c) 8 (d) 7
  15. 15.What is the smallest number you must add to 236 to make it divisible by 3? (a) 1 (b) 2
  16. 16.What is the units digit of 9^39? (a) 9 (b) 7 (c) 1 (d) 8
  17. 17.What is the remainder when 57 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
  18. 18.What is the smallest number you must add to 169 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 4
  19. 19.What is the smallest number you must add to 81 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  20. 20.What is the units digit of 3^43? (a) 1 (b) 7 (c) 3 (d) 9
  21. 21.What is the remainder when 88 is divided by 3? (a) 0 (b) 1 (c) 2
  22. 22.What is the units digit of 9^21? (a) 8 (b) 1 (c) 7 (d) 9
  23. 23.What is the smallest number you must add to 217 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  24. 24.What is the remainder when 197 is divided by 8? (a) 6 (b) 5 (c) 4 (d) 3
  25. 25.What is the smallest number you must add to 468 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 8
  26. 26.What is the remainder when 186 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 6
  27. 27.What is the remainder when 697 is divided by 10? (a) 8 (b) 7 (c) 6 (d) 3
  28. 28.What is the units digit of 8^11? (a) 8 (b) 2 (c) 4 (d) 6
  29. 29.What is the units digit of 7^45? (a) 7 (b) 1 (c) 3 (d) 9
  30. 30.Which of these numbers is divisible by 8? (a) 6666 (b) 6672 (c) 6682 (d) 6675
  31. 31.Which of these numbers is divisible by 3? (a) 1681 (b) 1678 (c) 1684 (d) 1680
  32. 32.Which of these numbers is divisible by 8? (a) 1076 (b) 1064 (c) 1071 (d) 1063
  33. 33.What is the remainder when 329 is divided by 8? (a) 1 (b) 7 (c) 2 (d) 0
  34. 34.What is the units digit of 2^58? (a) 2 (b) 6 (c) 4 (d) 8
  35. 35.What is the smallest number you must add to 472 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 1
  36. 36.What is the units digit of 7^15? (a) 9 (b) 1 (c) 7 (d) 3
  37. 37.What is the smallest number you must add to 718 to make it divisible by 12? (a) 3 (b) 2 (c) 10 (d) 1
  38. 38.What is the smallest number you must add to 515 to make it divisible by 11? (a) 3 (b) 2 (c) 1 (d) 9
  39. 39.Which of these numbers is divisible by 3? (a) 881 (b) 876 (c) 874 (d) 878
  40. 40.What is the smallest number you must add to 138 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2

Answer key

  1. 822
  2. 1
  3. 3
  4. 4
  5. 6030
  6. 1782
  7. 6
  8. 473
  9. 4
  10. 6
  11. 9
  12. 2079
  13. 4
  14. 7
  15. 1
  16. 9
  17. 2
  18. 5
  19. 4
  20. 7
  21. 1
  22. 9
  23. 3
  24. 5
  25. 2
  26. 4
  27. 7
  28. 2
  29. 7
  30. 6672
  31. 1680
  32. 1064
  33. 1
  34. 4
  35. 2
  36. 3
  37. 2
  38. 2
  39. 876
  40. 2

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