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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 403 · 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 346 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
  2. 2.Which of these numbers is divisible by 4? (a) 901 (b) 897 (c) 906 (d) 900
  3. 3.Which of these numbers is divisible by 9? (a) 8055 (b) 8067 (c) 8050 (d) 8063
  4. 4.What is the remainder when 82 is divided by 3? (a) 0 (b) 2 (c) 1
  5. 5.What is the smallest number you must add to 548 to make it divisible by 11? (a) 2 (b) 9 (c) 3 (d) 1
  6. 6.Which of these numbers is divisible by 6? (a) 1619 (b) 1611 (c) 1614 (d) 1621
  7. 7.What is the remainder when 804 is divided by 11? (a) 2 (b) 1 (c) 10 (d) 0
  8. 8.What is the remainder when 459 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
  9. 9.What is the remainder when 46 is divided by 3? (a) 2 (b) 1 (c) 0
  10. 10.What is the smallest number you must add to 203 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 5
  11. 11.What is the smallest number you must add to 63 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
  12. 12.What is the smallest number you must add to 202 to make it divisible by 3? (a) 1 (b) 2
  13. 13.Which of these numbers is divisible by 6? (a) 3637 (b) 3629 (c) 3630 (d) 3634
  14. 14.Which of these numbers is divisible by 9? (a) 4034 (b) 4041 (c) 4055 (d) 4044
  15. 15.What is the smallest number you must add to 383 to make it divisible by 11? (a) 3 (b) 9 (c) 1 (d) 2
  16. 16.Which of these numbers is divisible by 3? (a) 1018 (b) 1015 (c) 1013 (d) 1014
  17. 17.What is the smallest number you must add to 54 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  18. 18.Which of these numbers is divisible by 8? (a) 1466 (b) 1459 (c) 1477 (d) 1464
  19. 19.What is the smallest number you must add to 110 to make it divisible by 9? (a) 6 (b) 2 (c) 7 (d) 8
  20. 20.What is the smallest number you must add to 471 to make it divisible by 11? (a) 9 (b) 2 (c) 1 (d) 3
  21. 21.What is the units digit of 3^42? (a) 9 (b) 3 (c) 7 (d) 1
  22. 22.Which of these numbers is divisible by 6? (a) 2741 (b) 2732 (c) 2730 (d) 2726
  23. 23.What is the remainder when 493 is divided by 8? (a) 5 (b) 6 (c) 4 (d) 3
  24. 24.What is the remainder when 78 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
  25. 25.What is the units digit of 3^34? (a) 9 (b) 7 (c) 1 (d) 3
  26. 26.What is the smallest number you must add to 385 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
  27. 27.What is the remainder when 111 is divided by 8? (a) 7 (b) 5 (c) 1 (d) 6
  28. 28.What is the remainder when 101 is divided by 3? (a) 0 (b) 2 (c) 1
  29. 29.What is the smallest number you must add to 259 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  30. 30.Which of these numbers is divisible by 11? (a) 3958 (b) 3969 (c) 3973 (d) 3960
  31. 31.What is the units digit of 9^24? (a) 9 (b) 1 (c) 0 (d) 2
  32. 32.What is the smallest number you must add to 684 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
  33. 33.What is the remainder when 529 is divided by 9? (a) 7 (b) 2 (c) 8 (d) 6
  34. 34.What is the smallest number you must add to 536 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 3
  35. 35.Which of these numbers is divisible by 11? (a) 1872 (b) 1870 (c) 1861 (d) 1888
  36. 36.Which of these numbers is divisible by 3? (a) 2181 (b) 2180 (c) 2182 (d) 2185
  37. 37.Which of these numbers is divisible by 11? (a) 5848 (b) 5852 (c) 5868 (d) 5856
  38. 38.Which of these numbers is divisible by 9? (a) 2801 (b) 2792 (c) 2799 (d) 2810
  39. 39.Which of these numbers is divisible by 9? (a) 4887 (b) 4882 (c) 4893 (d) 4899
  40. 40.What is the remainder when 515 is divided by 11? (a) 10 (b) 2 (c) 8 (d) 9

Answer key

  1. 6
  2. 900
  3. 8055
  4. 1
  5. 2
  6. 1614
  7. 1
  8. 3
  9. 1
  10. 1
  11. 3
  12. 2
  13. 3630
  14. 4041
  15. 2
  16. 1014
  17. 1
  18. 1464
  19. 7
  20. 2
  21. 9
  22. 2730
  23. 5
  24. 3
  25. 9
  26. 5
  27. 7
  28. 2
  29. 1
  30. 3960
  31. 1
  32. 6
  33. 7
  34. 3
  35. 1870
  36. 2181
  37. 5852
  38. 2799
  39. 4887
  40. 9

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