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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 218 · 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) 2065 (b) 2064 (c) 2062 (d) 2068
  2. 2.What is the units digit of 3^21? (a) 3 (b) 1 (c) 7 (d) 9
  3. 3.What is the smallest number you must add to 117 to make it divisible by 8? (a) 4 (b) 2 (c) 5 (d) 3
  4. 4.What is the units digit of 9^21? (a) 8 (b) 9 (c) 1 (d) 7
  5. 5.Which of these numbers is divisible by 9? (a) 6625 (b) 6639 (c) 6623 (d) 6624
  6. 6.What is the smallest number you must add to 505 to make it divisible by 8? (a) 5 (b) 1 (c) 6 (d) 7
  7. 7.What is the remainder when 124 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 2
  8. 8.What is the remainder when 201 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
  9. 9.What is the smallest number you must add to 636 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
  10. 10.What is the remainder when 380 is divided by 6? (a) 3 (b) 2 (c) 1 (d) 4
  11. 11.What is the remainder when 436 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
  12. 12.Which of these numbers is divisible by 11? (a) 5565 (b) 5566 (c) 5580 (d) 5571
  13. 13.What is the remainder when 86 is divided by 7? (a) 2 (b) 3 (c) 1 (d) 5
  14. 14.What is the remainder when 654 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 7
  15. 15.Which of these numbers is divisible by 9? (a) 2934 (b) 2944 (c) 2930 (d) 2940
  16. 16.What is the smallest number you must add to 212 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  17. 17.Which of these numbers is divisible by 9? (a) 2085 (b) 2072 (c) 2079 (d) 2094
  18. 18.What is the smallest number you must add to 527 to make it divisible by 10? (a) 7 (b) 2 (c) 4 (d) 3
  19. 19.What is the smallest number you must add to 842 to make it divisible by 12? (a) 11 (b) 2 (c) 9 (d) 10
  20. 20.Which of these numbers is divisible by 6? (a) 409 (b) 402 (c) 405 (d) 400
  21. 21.What is the remainder when 523 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
  22. 22.Which of these numbers is divisible by 9? (a) 945 (b) 942 (c) 958 (d) 951
  23. 23.Which of these numbers is divisible by 11? (a) 9878 (b) 9885 (c) 9877 (d) 9896
  24. 24.What is the units digit of 8^27? (a) 6 (b) 4 (c) 2 (d) 8
  25. 25.Which of these numbers is divisible by 3? (a) 1556 (b) 1553 (c) 1551 (d) 1550
  26. 26.What is the smallest number you must add to 266 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  27. 27.What is the units digit of 8^39? (a) 4 (b) 6 (c) 2 (d) 8
  28. 28.What is the remainder when 139 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 1
  29. 29.What is the smallest number you must add to 467 to make it divisible by 9? (a) 3 (b) 8 (c) 1 (d) 2
  30. 30.What is the units digit of 9^54? (a) 9 (b) 1 (c) 0 (d) 2
  31. 31.Which of these numbers is divisible by 3? (a) 1824 (b) 1823 (c) 1825 (d) 1829
  32. 32.What is the remainder when 587 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 2
  33. 33.What is the units digit of 3^45? (a) 1 (b) 3 (c) 7 (d) 9
  34. 34.What is the smallest number you must add to 82 to make it divisible by 7? (a) 1 (b) 2 (c) 5 (d) 3
  35. 35.What is the smallest number you must add to 67 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  36. 36.What is the smallest number you must add to 58 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  37. 37.What is the remainder when 279 is divided by 12? (a) 3 (b) 4 (c) 9 (d) 2
  38. 38.Which of these numbers is divisible by 9? (a) 1588 (b) 1571 (c) 1575 (d) 1581
  39. 39.What is the smallest number you must add to 135 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  40. 40.What is the smallest number you must add to 658 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 1

Answer key

  1. 2064
  2. 3
  3. 3
  4. 9
  5. 6624
  6. 7
  7. 5
  8. 1
  9. 4
  10. 2
  11. 4
  12. 5566
  13. 2
  14. 6
  15. 2934
  16. 3
  17. 2079
  18. 3
  19. 10
  20. 402
  21. 5
  22. 945
  23. 9878
  24. 2
  25. 1551
  26. 4
  27. 2
  28. 6
  29. 1
  30. 1
  31. 1824
  32. 3
  33. 3
  34. 2
  35. 3
  36. 2
  37. 3
  38. 1575
  39. 1
  40. 8

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