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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 585 · 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 smallest number you must add to 381 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
  2. 2.What is the units digit of 9^21? (a) 9 (b) 7 (c) 8 (d) 1
  3. 3.What is the remainder when 377 is divided by 8? (a) 1 (b) 2 (c) 0 (d) 7
  4. 4.What is the remainder when 302 is divided by 12? (a) 1 (b) 2 (c) 3 (d) 10
  5. 5.What is the remainder when 337 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
  6. 6.What is the smallest number you must add to 503 to make it divisible by 7? (a) 6 (b) 3 (c) 2 (d) 1
  7. 7.What is the smallest number you must add to 135 to make it divisible by 10? (a) 7 (b) 5 (c) 6 (d) 4
  8. 8.What is the smallest number you must add to 306 to make it divisible by 8? (a) 6 (b) 7 (c) 2 (d) 5
  9. 9.What is the smallest number you must add to 311 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  10. 10.What is the smallest number you must add to 157 to make it divisible by 3? (a) 1 (b) 2
  11. 11.What is the remainder when 313 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  12. 12.What is the smallest number you must add to 69 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
  13. 13.Which of these numbers is divisible by 4? (a) 869 (b) 875 (c) 868 (d) 865
  14. 14.What is the remainder when 127 is divided by 8? (a) 6 (b) 1 (c) 7 (d) 5
  15. 15.What is the remainder when 725 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
  16. 16.Which of these numbers is divisible by 11? (a) 4286 (b) 4279 (c) 4276 (d) 4298
  17. 17.What is the smallest number you must add to 145 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
  18. 18.What is the smallest number you must add to 166 to make it divisible by 3? (a) 1 (b) 2
  19. 19.What is the units digit of 7^21? (a) 3 (b) 7 (c) 1 (d) 9
  20. 20.What is the remainder when 188 is divided by 12? (a) 4 (b) 9 (c) 8 (d) 7
  21. 21.What is the remainder when 151 is divided by 12? (a) 8 (b) 6 (c) 5 (d) 7
  22. 22.What is the smallest number you must add to 416 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5
  23. 23.Which of these numbers is divisible by 3? (a) 298 (b) 294 (c) 292 (d) 296
  24. 24.Which of these numbers is divisible by 6? (a) 569 (b) 570 (c) 574 (d) 578
  25. 25.Which of these numbers is divisible by 3? (a) 1183 (b) 1181 (c) 1179 (d) 1177
  26. 26.What is the units digit of 4^21? (a) 5 (b) 4 (c) 3 (d) 6
  27. 27.Which of these numbers is divisible by 11? (a) 8140 (b) 8137 (c) 8150 (d) 8156
  28. 28.What is the remainder when 628 is divided by 11? (a) 0 (b) 10 (c) 1 (d) 2
  29. 29.What is the remainder when 616 is divided by 12? (a) 4 (b) 8 (c) 5 (d) 3
  30. 30.What is the remainder when 201 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
  31. 31.What is the units digit of 2^26? (a) 2 (b) 8 (c) 6 (d) 4
  32. 32.What is the units digit of 8^36? (a) 2 (b) 4 (c) 6 (d) 8
  33. 33.What is the remainder when 241 is divided by 3? (a) 0 (b) 1 (c) 2
  34. 34.What is the units digit of 2^46? (a) 6 (b) 4 (c) 2 (d) 8
  35. 35.What is the smallest number you must add to 424 to make it divisible by 9? (a) 1 (b) 6 (c) 8 (d) 7
  36. 36.What is the smallest number you must add to 478 to make it divisible by 9? (a) 8 (b) 6 (c) 1 (d) 7
  37. 37.Which of these numbers is divisible by 4? (a) 786 (b) 790 (c) 781 (d) 784
  38. 38.Which of these numbers is divisible by 8? (a) 807 (b) 797 (c) 800 (d) 809
  39. 39.What is the remainder when 70 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  40. 40.Which of these numbers is divisible by 6? (a) 4921 (b) 4920 (c) 4917 (d) 4927

Answer key

  1. 6
  2. 9
  3. 1
  4. 2
  5. 2
  6. 1
  7. 5
  8. 6
  9. 4
  10. 2
  11. 1
  12. 1
  13. 868
  14. 7
  15. 5
  16. 4279
  17. 5
  18. 2
  19. 7
  20. 8
  21. 7
  22. 4
  23. 294
  24. 570
  25. 1179
  26. 4
  27. 8140
  28. 1
  29. 4
  30. 1
  31. 4
  32. 6
  33. 1
  34. 4
  35. 8
  36. 8
  37. 784
  38. 800
  39. 2
  40. 4920

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