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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 158 · 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 622 is divided by 9? (a) 1 (b) 8 (c) 0 (d) 2
  2. 2.What is the units digit of 7^15? (a) 9 (b) 3 (c) 7 (d) 1
  3. 3.What is the smallest number you must add to 183 to make it divisible by 9? (a) 3 (b) 7 (c) 6 (d) 5
  4. 4.What is the remainder when 469 is divided by 12? (a) 2 (b) 11 (c) 0 (d) 1
  5. 5.What is the remainder when 510 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 7
  6. 6.What is the remainder when 357 is divided by 11? (a) 4 (b) 7 (c) 5 (d) 6
  7. 7.Which of these numbers is divisible by 11? (a) 1824 (b) 1814 (c) 1815 (d) 1831
  8. 8.What is the smallest number you must add to 299 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  9. 9.What is the smallest number you must add to 719 to make it divisible by 11? (a) 8 (b) 4 (c) 7 (d) 6
  10. 10.What is the units digit of 3^47? (a) 7 (b) 3 (c) 9 (d) 1
  11. 11.Which of these numbers is divisible by 3? (a) 404 (b) 405 (c) 406 (d) 409
  12. 12.What is the smallest number you must add to 279 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  13. 13.What is the smallest number you must add to 721 to make it divisible by 10? (a) 9 (b) 7 (c) 8 (d) 1
  14. 14.What is the smallest number you must add to 71 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  15. 15.What is the remainder when 157 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
  16. 16.Which of these numbers is divisible by 8? (a) 6564 (b) 6575 (c) 6582 (d) 6568
  17. 17.What is the smallest number you must add to 149 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 5
  18. 18.What is the remainder when 529 is divided by 11? (a) 0 (b) 2 (c) 1 (d) 10
  19. 19.What is the remainder when 711 is divided by 10? (a) 9 (b) 1 (c) 2 (d) 0
  20. 20.What is the smallest number you must add to 385 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
  21. 21.What is the remainder when 414 is divided by 11? (a) 4 (b) 7 (c) 8 (d) 6
  22. 22.Which of these numbers is divisible by 9? (a) 7209 (b) 7204 (c) 7216 (d) 7225
  23. 23.Which of these numbers is divisible by 8? (a) 1353 (b) 1344 (c) 1342 (d) 1346
  24. 24.Which of these numbers is divisible by 9? (a) 7962 (b) 7954 (c) 7946 (d) 7947
  25. 25.What is the smallest number you must add to 245 to make it divisible by 11? (a) 9 (b) 3 (c) 8 (d) 7
  26. 26.What is the remainder when 861 is divided by 11? (a) 2 (b) 3 (c) 8 (d) 4
  27. 27.What is the remainder when 558 is divided by 10? (a) 2 (b) 9 (c) 7 (d) 8
  28. 28.What is the smallest number you must add to 246 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 3
  29. 29.What is the units digit of 3^48? (a) 1 (b) 9 (c) 7 (d) 3
  30. 30.Which of these numbers is divisible by 4? (a) 3009 (b) 3008 (c) 3005 (d) 3014
  31. 31.What is the remainder when 152 is divided by 3? (a) 0 (b) 2 (c) 1
  32. 32.Which of these numbers is divisible by 3? (a) 1018 (b) 1013 (c) 1015 (d) 1014
  33. 33.Which of these numbers is divisible by 11? (a) 3675 (b) 3697 (c) 3693 (d) 3685
  34. 34.Which of these numbers is divisible by 9? (a) 3943 (b) 3935 (c) 3942 (d) 3957
  35. 35.What is the remainder when 287 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
  36. 36.What is the smallest number you must add to 73 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  37. 37.What is the units digit of 8^48? (a) 4 (b) 8 (c) 2 (d) 6
  38. 38.What is the remainder when 51 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
  39. 39.What is the smallest number you must add to 396 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  40. 40.What is the units digit of 4^51? (a) 3 (b) 5 (c) 4 (d) 6

Answer key

  1. 1
  2. 3
  3. 6
  4. 1
  5. 6
  6. 5
  7. 1815
  8. 1
  9. 7
  10. 7
  11. 405
  12. 1
  13. 9
  14. 1
  15. 1
  16. 6568
  17. 1
  18. 1
  19. 1
  20. 5
  21. 7
  22. 7209
  23. 1344
  24. 7947
  25. 8
  26. 3
  27. 8
  28. 4
  29. 1
  30. 3008
  31. 2
  32. 1014
  33. 3685
  34. 3942
  35. 3
  36. 3
  37. 6
  38. 1
  39. 4
  40. 4

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