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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 583 · 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 407 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
  2. 2.What is the remainder when 205 is divided by 3? (a) 2 (b) 0 (c) 1
  3. 3.What is the remainder when 475 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 1
  4. 4.What is the remainder when 143 is divided by 6? (a) 1 (b) 5 (c) 4 (d) 3
  5. 5.What is the smallest number you must add to 311 to make it divisible by 9? (a) 3 (b) 4 (c) 5 (d) 6
  6. 6.What is the smallest number you must add to 675 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
  7. 7.Which of these numbers is divisible by 3? (a) 1729 (b) 1736 (c) 1731 (d) 1732
  8. 8.Which of these numbers is divisible by 11? (a) 1173 (b) 1166 (c) 1164 (d) 1185
  9. 9.What is the smallest number you must add to 71 to make it divisible by 3? (a) 1 (b) 2
  10. 10.What is the units digit of 3^14? (a) 1 (b) 7 (c) 3 (d) 9
  11. 11.What is the remainder when 392 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 8
  12. 12.Which of these numbers is divisible by 11? (a) 3290 (b) 3287 (c) 3272 (d) 3278
  13. 13.What is the smallest number you must add to 106 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  14. 14.What is the smallest number you must add to 62 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
  15. 15.What is the units digit of 7^54? (a) 9 (b) 1 (c) 3 (d) 7
  16. 16.What is the units digit of 2^33? (a) 6 (b) 2 (c) 4 (d) 8
  17. 17.What is the smallest number you must add to 405 to make it divisible by 8? (a) 4 (b) 2 (c) 5 (d) 3
  18. 18.What is the units digit of 7^38? (a) 9 (b) 1 (c) 7 (d) 3
  19. 19.What is the remainder when 309 is divided by 9? (a) 3 (b) 4 (c) 2 (d) 6
  20. 20.What is the smallest number you must add to 157 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  21. 21.What is the units digit of 2^42? (a) 8 (b) 6 (c) 4 (d) 2
  22. 22.What is the units digit of 2^41? (a) 6 (b) 8 (c) 4 (d) 2
  23. 23.What is the smallest number you must add to 667 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 1
  24. 24.What is the smallest number you must add to 116 to make it divisible by 3? (a) 2 (b) 1
  25. 25.Which of these numbers is divisible by 9? (a) 1412 (b) 1429 (c) 1413 (d) 1415
  26. 26.What is the remainder when 425 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
  27. 27.What is the units digit of 2^40? (a) 2 (b) 4 (c) 8 (d) 6
  28. 28.Which of these numbers is divisible by 8? (a) 4000 (b) 4002 (c) 4011 (d) 3994
  29. 29.Which of these numbers is divisible by 11? (a) 7470 (b) 7458 (c) 7468 (d) 7448
  30. 30.What is the remainder when 227 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
  31. 31.What is the remainder when 589 is divided by 8? (a) 3 (b) 4 (c) 5 (d) 6
  32. 32.What is the remainder when 295 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
  33. 33.What is the remainder when 721 is divided by 12? (a) 0 (b) 1 (c) 2 (d) 11
  34. 34.What is the remainder when 117 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
  35. 35.What is the remainder when 142 is divided by 10? (a) 8 (b) 2 (c) 3 (d) 1
  36. 36.What is the smallest number you must add to 133 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  37. 37.What is the units digit of 2^35? (a) 2 (b) 8 (c) 4 (d) 6
  38. 38.Which of these numbers is divisible by 3? (a) 1437 (b) 1439 (c) 1436 (d) 1441
  39. 39.What is the remainder when 331 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
  40. 40.What is the smallest number you must add to 631 to make it divisible by 11? (a) 6 (b) 4 (c) 8 (d) 7

Answer key

  1. 1
  2. 1
  3. 6
  4. 5
  5. 4
  6. 5
  7. 1731
  8. 1166
  9. 1
  10. 9
  11. 7
  12. 3278
  13. 2
  14. 3
  15. 9
  16. 2
  17. 3
  18. 9
  19. 3
  20. 3
  21. 4
  22. 2
  23. 8
  24. 1
  25. 1413
  26. 5
  27. 6
  28. 4000
  29. 7458
  30. 2
  31. 5
  32. 3
  33. 1
  34. 1
  35. 2
  36. 3
  37. 8
  38. 1437
  39. 1
  40. 7

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