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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 451 · 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 173 is divided by 3? (a) 0 (b) 2 (c) 1
  2. 2.What is the remainder when 415 is divided by 11? (a) 8 (b) 9 (c) 7 (d) 3
  3. 3.Which of these numbers is divisible by 4? (a) 1344 (b) 1345 (c) 1351 (d) 1341
  4. 4.What is the remainder when 187 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 2
  5. 5.Which of these numbers is divisible by 11? (a) 5840 (b) 5844 (c) 5841 (d) 5860
  6. 6.Which of these numbers is divisible by 11? (a) 2068 (b) 2072 (c) 2080 (d) 2063
  7. 7.Which of these numbers is divisible by 4? (a) 309 (b) 308 (c) 314 (d) 306
  8. 8.What is the units digit of 9^22? (a) 2 (b) 1 (c) 9 (d) 0
  9. 9.Which of these numbers is divisible by 9? (a) 4286 (b) 4295 (c) 4304 (d) 4293
  10. 10.What is the smallest number you must add to 287 to make it divisible by 12? (a) 1 (b) 11 (c) 2 (d) 3
  11. 11.What is the remainder when 211 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
  12. 12.Which of these numbers is divisible by 9? (a) 405 (b) 418 (c) 401 (d) 406
  13. 13.What is the smallest number you must add to 459 to make it divisible by 10? (a) 1 (b) 2 (c) 3 (d) 9
  14. 14.Which of these numbers is divisible by 4? (a) 2136 (b) 2134 (c) 2143 (d) 2138
  15. 15.What is the smallest number you must add to 105 to make it divisible by 9? (a) 3 (b) 6 (c) 4 (d) 2
  16. 16.Which of these numbers is divisible by 4? (a) 1032 (b) 1031 (c) 1034 (d) 1037
  17. 17.Which of these numbers is divisible by 9? (a) 6290 (b) 6294 (c) 6291 (d) 6306
  18. 18.What is the smallest number you must add to 158 to make it divisible by 3? (a) 2 (b) 1
  19. 19.Which of these numbers is divisible by 3? (a) 1985 (b) 1981 (c) 1979 (d) 1980
  20. 20.What is the units digit of 8^41? (a) 6 (b) 2 (c) 8 (d) 4
  21. 21.Which of these numbers is divisible by 6? (a) 2411 (b) 2403 (c) 2400 (d) 2399
  22. 22.What is the smallest number you must add to 184 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
  23. 23.Which of these numbers is divisible by 9? (a) 6579 (b) 6589 (c) 6583 (d) 6577
  24. 24.What is the remainder when 307 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
  25. 25.What is the remainder when 232 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6
  26. 26.What is the smallest number you must add to 182 to make it divisible by 9? (a) 2 (b) 6 (c) 7 (d) 8
  27. 27.What is the smallest number you must add to 230 to make it divisible by 8? (a) 6 (b) 2 (c) 1 (d) 3
  28. 28.Which of these numbers is divisible by 3? (a) 922 (b) 921 (c) 925 (d) 919
  29. 29.Which of these numbers is divisible by 9? (a) 661 (b) 679 (c) 670 (d) 666
  30. 30.What is the units digit of 8^58? (a) 8 (b) 4 (c) 2 (d) 6
  31. 31.What is the smallest number you must add to 323 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 8
  32. 32.What is the smallest number you must add to 88 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
  33. 33.Which of these numbers is divisible by 3? (a) 1652 (b) 1648 (c) 1645 (d) 1647
  34. 34.What is the remainder when 483 is divided by 11? (a) 10 (b) 1 (c) 8 (d) 9
  35. 35.Which of these numbers is divisible by 9? (a) 6359 (b) 6349 (c) 6354 (d) 6364
  36. 36.Which of these numbers is divisible by 11? (a) 4069 (b) 4084 (c) 4070 (d) 4073
  37. 37.What is the remainder when 715 is divided by 12? (a) 6 (b) 5 (c) 8 (d) 7
  38. 38.Which of these numbers is divisible by 9? (a) 7328 (b) 7335 (c) 7337 (d) 7351
  39. 39.What is the remainder when 634 is divided by 10? (a) 5 (b) 3 (c) 4 (d) 6
  40. 40.What is the smallest number you must add to 165 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6

Answer key

  1. 2
  2. 8
  3. 1344
  4. 3
  5. 5841
  6. 2068
  7. 308
  8. 1
  9. 4293
  10. 1
  11. 1
  12. 405
  13. 1
  14. 2136
  15. 3
  16. 1032
  17. 6291
  18. 1
  19. 1980
  20. 8
  21. 2400
  22. 2
  23. 6579
  24. 2
  25. 1
  26. 7
  27. 2
  28. 921
  29. 666
  30. 4
  31. 7
  32. 3
  33. 1647
  34. 10
  35. 6354
  36. 4070
  37. 7
  38. 7335
  39. 4
  40. 5

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