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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 288 · 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 349 to make it divisible by 9? (a) 7 (b) 3 (c) 2 (d) 1
  2. 2.Which of these numbers is divisible by 6? (a) 4636 (b) 4624 (c) 4628 (d) 4626
  3. 3.What is the remainder when 412 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
  4. 4.What is the smallest number you must add to 135 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  5. 5.What is the smallest number you must add to 294 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  6. 6.What is the smallest number you must add to 334 to make it divisible by 10? (a) 4 (b) 7 (c) 6 (d) 5
  7. 7.Which of these numbers is divisible by 4? (a) 2008 (b) 2005 (c) 2014 (d) 2010
  8. 8.What is the units digit of 3^16? (a) 3 (b) 7 (c) 1 (d) 9
  9. 9.What is the smallest number you must add to 207 to make it divisible by 7? (a) 3 (b) 5 (c) 2 (d) 4
  10. 10.Which of these numbers is divisible by 11? (a) 7990 (b) 7986 (c) 7981 (d) 8004
  11. 11.What is the smallest number you must add to 308 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
  12. 12.What is the smallest number you must add to 288 to make it divisible by 10? (a) 2 (b) 8 (c) 3 (d) 1
  13. 13.Which of these numbers is divisible by 9? (a) 5437 (b) 5427 (c) 5426 (d) 5433
  14. 14.What is the units digit of 3^24? (a) 1 (b) 3 (c) 9 (d) 7
  15. 15.Which of these numbers is divisible by 9? (a) 6804 (b) 6816 (c) 6806 (d) 6800
  16. 16.What is the remainder when 455 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 1
  17. 17.Which of these numbers is divisible by 11? (a) 7268 (b) 7284 (c) 7278 (d) 7271
  18. 18.What is the remainder when 619 is divided by 11? (a) 2 (b) 8 (c) 3 (d) 4
  19. 19.What is the remainder when 369 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
  20. 20.What is the units digit of 3^55? (a) 1 (b) 9 (c) 3 (d) 7
  21. 21.What is the smallest number you must add to 65 to make it divisible by 3? (a) 1 (b) 2
  22. 22.What is the smallest number you must add to 125 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  23. 23.Which of these numbers is divisible by 9? (a) 7845 (b) 7848 (c) 7861 (d) 7850
  24. 24.What is the units digit of 4^13? (a) 5 (b) 6 (c) 4 (d) 3
  25. 25.What is the units digit of 9^28? (a) 2 (b) 9 (c) 0 (d) 1
  26. 26.What is the remainder when 574 is divided by 8? (a) 2 (b) 5 (c) 7 (d) 6
  27. 27.Which of these numbers is divisible by 8? (a) 3016 (b) 3031 (c) 3015 (d) 3019
  28. 28.What is the smallest number you must add to 507 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 3
  29. 29.What is the remainder when 187 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5
  30. 30.What is the smallest number you must add to 635 to make it divisible by 9? (a) 5 (b) 3 (c) 6 (d) 4
  31. 31.What is the remainder when 285 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
  32. 32.Which of these numbers is divisible by 4? (a) 2238 (b) 2231 (c) 2232 (d) 2235
  33. 33.What is the smallest number you must add to 403 to make it divisible by 6? (a) 1 (b) 5 (c) 4 (d) 3
  34. 34.What is the smallest number you must add to 631 to make it divisible by 11? (a) 4 (b) 8 (c) 6 (d) 7
  35. 35.What is the remainder when 424 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 8
  36. 36.Which of these numbers is divisible by 4? (a) 880 (b) 883 (c) 879 (d) 885
  37. 37.Which of these numbers is divisible by 11? (a) 2671 (b) 2659 (c) 2662 (d) 2682
  38. 38.What is the units digit of 3^25? (a) 3 (b) 1 (c) 7 (d) 9
  39. 39.What is the smallest number you must add to 428 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
  40. 40.What is the smallest number you must add to 601 to make it divisible by 12? (a) 11 (b) 1 (c) 10 (d) 9

Answer key

  1. 2
  2. 4626
  3. 2
  4. 1
  5. 2
  6. 6
  7. 2008
  8. 1
  9. 3
  10. 7986
  11. 4
  12. 2
  13. 5427
  14. 1
  15. 6804
  16. 5
  17. 7271
  18. 3
  19. 3
  20. 7
  21. 1
  22. 3
  23. 7848
  24. 4
  25. 1
  26. 6
  27. 3016
  28. 4
  29. 5
  30. 4
  31. 1
  32. 2232
  33. 5
  34. 7
  35. 6
  36. 880
  37. 2662
  38. 3
  39. 6
  40. 11

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