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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 834 · 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 4? (a) 2637 (b) 2631 (c) 2632 (d) 2635
  2. 2.What is the units digit of 2^20? (a) 6 (b) 8 (c) 2 (d) 4
  3. 3.What is the smallest number you must add to 89 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 1
  4. 4.Which of these numbers is divisible by 3? (a) 676 (b) 683 (c) 679 (d) 678
  5. 5.What is the smallest number you must add to 132 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
  6. 6.What is the remainder when 246 is divided by 8? (a) 6 (b) 7 (c) 5 (d) 2
  7. 7.What is the smallest number you must add to 260 to make it divisible by 12? (a) 8 (b) 5 (c) 4 (d) 3
  8. 8.Which of these numbers is divisible by 3? (a) 2218 (b) 2217 (c) 2221 (d) 2215
  9. 9.What is the smallest number you must add to 136 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 1
  10. 10.Which of these numbers is divisible by 9? (a) 3667 (b) 3661 (c) 3653 (d) 3654
  11. 11.What is the units digit of 2^51? (a) 6 (b) 4 (c) 8 (d) 2
  12. 12.What is the remainder when 391 is divided by 8? (a) 5 (b) 6 (c) 7 (d) 1
  13. 13.Which of these numbers is divisible by 11? (a) 4407 (b) 4412 (c) 4393 (d) 4400
  14. 14.What is the smallest number you must add to 182 to make it divisible by 3? (a) 1 (b) 2
  15. 15.What is the smallest number you must add to 515 to make it divisible by 12? (a) 2 (b) 3 (c) 1 (d) 11
  16. 16.What is the remainder when 89 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
  17. 17.Which of these numbers is divisible by 4? (a) 1953 (b) 1951 (c) 1959 (d) 1952
  18. 18.Which of these numbers is divisible by 9? (a) 8037 (b) 8044 (c) 8034 (d) 8054
  19. 19.What is the remainder when 58 is divided by 3? (a) 2 (b) 1 (c) 0
  20. 20.Which of these numbers is divisible by 9? (a) 6125 (b) 6120 (c) 6132 (d) 6118
  21. 21.Which of these numbers is divisible by 9? (a) 6377 (b) 6366 (c) 6359 (d) 6363
  22. 22.What is the remainder when 334 is divided by 9? (a) 2 (b) 0 (c) 8 (d) 1
  23. 23.Which of these numbers is divisible by 4? (a) 870 (b) 865 (c) 873 (d) 868
  24. 24.What is the remainder when 184 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
  25. 25.What is the smallest number you must add to 277 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  26. 26.What is the remainder when 211 is divided by 3? (a) 2 (b) 0 (c) 1
  27. 27.What is the smallest number you must add to 154 to make it divisible by 3? (a) 1 (b) 2
  28. 28.Which of these numbers is divisible by 6? (a) 561 (b) 566 (c) 558 (d) 554
  29. 29.What is the smallest number you must add to 222 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  30. 30.What is the smallest number you must add to 267 to make it divisible by 11? (a) 7 (b) 8 (c) 9 (d) 3
  31. 31.What is the units digit of 7^58? (a) 7 (b) 9 (c) 3 (d) 1
  32. 32.What is the units digit of 4^27? (a) 4 (b) 6 (c) 3 (d) 5
  33. 33.What is the remainder when 399 is divided by 8? (a) 1 (b) 5 (c) 7 (d) 6
  34. 34.Which of these numbers is divisible by 8? (a) 1470 (b) 1464 (c) 1460 (d) 1476
  35. 35.Which of these numbers is divisible by 6? (a) 3976 (b) 3978 (c) 3981 (d) 3987
  36. 36.What is the smallest number you must add to 511 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 1
  37. 37.What is the units digit of 9^50? (a) 1 (b) 9 (c) 2 (d) 0
  38. 38.Which of these numbers is divisible by 9? (a) 2720 (b) 2707 (c) 2709 (d) 2711
  39. 39.Which of these numbers is divisible by 4? (a) 784 (b) 782 (c) 790 (d) 785
  40. 40.What is the remainder when 502 is divided by 11? (a) 4 (b) 8 (c) 6 (d) 7

Answer key

  1. 2632
  2. 6
  3. 7
  4. 678
  5. 4
  6. 6
  7. 4
  8. 2217
  9. 2
  10. 3654
  11. 8
  12. 7
  13. 4400
  14. 1
  15. 1
  16. 1
  17. 1952
  18. 8037
  19. 1
  20. 6120
  21. 6363
  22. 1
  23. 868
  24. 4
  25. 3
  26. 1
  27. 2
  28. 558
  29. 2
  30. 8
  31. 9
  32. 4
  33. 7
  34. 1464
  35. 3978
  36. 9
  37. 1
  38. 2709
  39. 784
  40. 7

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