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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 325 · 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 565 is divided by 9? (a) 6 (b) 8 (c) 2 (d) 7
  2. 2.What is the smallest number you must add to 89 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  3. 3.What is the smallest number you must add to 495 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 7
  4. 4.What is the remainder when 277 is divided by 7? (a) 5 (b) 6 (c) 3 (d) 4
  5. 5.What is the smallest number you must add to 135 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
  6. 6.What is the remainder when 31 is divided by 3? (a) 1 (b) 2 (c) 0
  7. 7.What is the smallest number you must add to 807 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 4
  8. 8.Which of these numbers is divisible by 4? (a) 963 (b) 960 (c) 966 (d) 957
  9. 9.Which of these numbers is divisible by 4? (a) 570 (b) 577 (c) 572 (d) 574
  10. 10.What is the units digit of 9^14? (a) 1 (b) 9 (c) 0 (d) 2
  11. 11.What is the remainder when 185 is divided by 8? (a) 1 (b) 0 (c) 7 (d) 2
  12. 12.What is the smallest number you must add to 81 to make it divisible by 8? (a) 5 (b) 1 (c) 7 (d) 6
  13. 13.What is the smallest number you must add to 468 to make it divisible by 11? (a) 4 (b) 5 (c) 6 (d) 7
  14. 14.What is the units digit of 7^56? (a) 3 (b) 1 (c) 9 (d) 7
  15. 15.Which of these numbers is divisible by 3? (a) 646 (b) 650 (c) 648 (d) 652
  16. 16.What is the units digit of 8^30? (a) 6 (b) 4 (c) 2 (d) 8
  17. 17.What is the smallest number you must add to 493 to make it divisible by 8? (a) 2 (b) 5 (c) 3 (d) 4
  18. 18.Which of these numbers is divisible by 8? (a) 420 (b) 416 (c) 409 (d) 428
  19. 19.Which of these numbers is divisible by 3? (a) 727 (b) 731 (c) 725 (d) 726
  20. 20.What is the units digit of 4^46? (a) 4 (b) 5 (c) 7 (d) 6
  21. 21.What is the smallest number you must add to 963 to make it divisible by 12? (a) 8 (b) 9 (c) 3 (d) 10
  22. 22.What is the remainder when 95 is divided by 3? (a) 2 (b) 0 (c) 1
  23. 23.What is the remainder when 455 is divided by 8? (a) 5 (b) 6 (c) 1 (d) 7
  24. 24.What is the units digit of 7^48? (a) 9 (b) 1 (c) 3 (d) 7
  25. 25.What is the smallest number you must add to 73 to make it divisible by 3? (a) 1 (b) 2
  26. 26.What is the smallest number you must add to 188 to make it divisible by 3? (a) 2 (b) 1
  27. 27.What is the remainder when 744 is divided by 11? (a) 8 (b) 7 (c) 6 (d) 4
  28. 28.Which of these numbers is divisible by 11? (a) 8294 (b) 8306 (c) 8301 (d) 8289
  29. 29.What is the smallest number you must add to 464 to make it divisible by 12? (a) 3 (b) 8 (c) 5 (d) 4
  30. 30.What is the smallest number you must add to 772 to make it divisible by 10? (a) 8 (b) 9 (c) 7 (d) 2
  31. 31.What is the units digit of 3^37? (a) 9 (b) 7 (c) 1 (d) 3
  32. 32.What is the units digit of 2^27? (a) 8 (b) 4 (c) 2 (d) 6
  33. 33.What is the remainder when 645 is divided by 11? (a) 7 (b) 4 (c) 6 (d) 8
  34. 34.What is the units digit of 4^54? (a) 6 (b) 4 (c) 7 (d) 5
  35. 35.What is the smallest number you must add to 232 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
  36. 36.Which of these numbers is divisible by 8? (a) 914 (b) 912 (c) 905 (d) 921
  37. 37.Which of these numbers is divisible by 11? (a) 3291 (b) 3287 (c) 3305 (d) 3289
  38. 38.What is the units digit of 3^15? (a) 1 (b) 9 (c) 7 (d) 3
  39. 39.Which of these numbers is divisible by 6? (a) 3848 (b) 3842 (c) 3840 (d) 3837
  40. 40.What is the smallest number you must add to 452 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 2

Answer key

  1. 7
  2. 3
  3. 5
  4. 4
  5. 3
  6. 1
  7. 7
  8. 960
  9. 572
  10. 1
  11. 1
  12. 7
  13. 5
  14. 1
  15. 648
  16. 4
  17. 3
  18. 416
  19. 726
  20. 6
  21. 9
  22. 2
  23. 7
  24. 1
  25. 2
  26. 1
  27. 7
  28. 8294
  29. 4
  30. 8
  31. 3
  32. 8
  33. 7
  34. 6
  35. 3
  36. 912
  37. 3289
  38. 7
  39. 3840
  40. 3

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