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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 69 · 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 units digit of 9^57? (a) 7 (b) 8 (c) 1 (d) 9
  2. 2.Which of these numbers is divisible by 8? (a) 4573 (b) 4560 (c) 4561 (d) 4557
  3. 3.What is the remainder when 321 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
  4. 4.What is the smallest number you must add to 95 to make it divisible by 8? (a) 3 (b) 1 (c) 2 (d) 7
  5. 5.Which of these numbers is divisible by 9? (a) 3129 (b) 3119 (c) 3114 (d) 3108
  6. 6.What is the remainder when 183 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
  7. 7.What is the remainder when 392 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2
  8. 8.Which of these numbers is divisible by 11? (a) 4083 (b) 4081 (c) 4078 (d) 4095
  9. 9.Which of these numbers is divisible by 4? (a) 3389 (b) 3382 (c) 3384 (d) 3386
  10. 10.Which of these numbers is divisible by 9? (a) 7134 (b) 7126 (c) 7144 (d) 7128
  11. 11.What is the smallest number you must add to 127 to make it divisible by 9? (a) 1 (b) 6 (c) 7 (d) 8
  12. 12.What is the remainder when 268 is divided by 9? (a) 2 (b) 6 (c) 8 (d) 7
  13. 13.What is the units digit of 4^27? (a) 6 (b) 4 (c) 3 (d) 5
  14. 14.Which of these numbers is divisible by 3? (a) 861 (b) 865 (c) 859 (d) 863
  15. 15.What is the smallest number you must add to 98 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
  16. 16.What is the remainder when 471 is divided by 11? (a) 9 (b) 2 (c) 8 (d) 10
  17. 17.What is the smallest number you must add to 506 to make it divisible by 8? (a) 6 (b) 2 (c) 7 (d) 5
  18. 18.What is the remainder when 429 is divided by 12? (a) 8 (b) 3 (c) 10 (d) 9
  19. 19.What is the smallest number you must add to 206 to make it divisible by 9? (a) 1 (b) 3 (c) 8 (d) 2
  20. 20.What is the remainder when 576 is divided by 10? (a) 6 (b) 7 (c) 4 (d) 5
  21. 21.What is the smallest number you must add to 49 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  22. 22.What is the remainder when 371 is divided by 6? (a) 5 (b) 1 (c) 4 (d) 3
  23. 23.What is the units digit of 9^36? (a) 9 (b) 1 (c) 2 (d) 0
  24. 24.Which of these numbers is divisible by 3? (a) 792 (b) 797 (c) 790 (d) 794
  25. 25.What is the smallest number you must add to 550 to make it divisible by 9? (a) 6 (b) 1 (c) 8 (d) 7
  26. 26.Which of these numbers is divisible by 4? (a) 1798 (b) 1796 (c) 1795 (d) 1803
  27. 27.What is the smallest number you must add to 348 to make it divisible by 8? (a) 5 (b) 6 (c) 3 (d) 4
  28. 28.What is the smallest number you must add to 943 to make it divisible by 12? (a) 5 (b) 6 (c) 4 (d) 7
  29. 29.What is the units digit of 2^30? (a) 6 (b) 8 (c) 4 (d) 2
  30. 30.What is the smallest number you must add to 620 to make it divisible by 9? (a) 8 (b) 2 (c) 1 (d) 3
  31. 31.Which of these numbers is divisible by 8? (a) 4456 (b) 4469 (c) 4460 (d) 4449
  32. 32.What is the remainder when 378 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  33. 33.What is the smallest number you must add to 549 to make it divisible by 10? (a) 2 (b) 1 (c) 9 (d) 3
  34. 34.What is the smallest number you must add to 890 to make it divisible by 12? (a) 2 (b) 11 (c) 10 (d) 9
  35. 35.What is the smallest number you must add to 229 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
  36. 36.What is the smallest number you must add to 150 to make it divisible by 7? (a) 4 (b) 3 (c) 5 (d) 6
  37. 37.What is the remainder when 397 is divided by 7? (a) 6 (b) 2 (c) 5 (d) 4
  38. 38.What is the smallest number you must add to 286 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 2
  39. 39.Which of these numbers is divisible by 4? (a) 758 (b) 754 (c) 761 (d) 756
  40. 40.Which of these numbers is divisible by 6? (a) 2585 (b) 2586 (c) 2588 (d) 2595

Answer key

  1. 9
  2. 4560
  3. 3
  4. 1
  5. 3114
  6. 3
  7. 2
  8. 4081
  9. 3384
  10. 7128
  11. 8
  12. 7
  13. 4
  14. 861
  15. 4
  16. 9
  17. 6
  18. 9
  19. 1
  20. 6
  21. 3
  22. 5
  23. 1
  24. 792
  25. 8
  26. 1796
  27. 4
  28. 5
  29. 4
  30. 1
  31. 4456
  32. 3
  33. 1
  34. 10
  35. 5
  36. 4
  37. 5
  38. 2
  39. 756
  40. 2586

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