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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 655 · 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.Which of these numbers is divisible by 4? (a) 3189 (b) 3188 (c) 3185 (d) 3194
  2. 2.What is the smallest number you must add to 353 to make it divisible by 10? (a) 8 (b) 6 (c) 3 (d) 7
  3. 3.What is the units digit of 2^10? (a) 8 (b) 4 (c) 2 (d) 6
  4. 4.What is the smallest number you must add to 611 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
  5. 5.Which of these numbers is divisible by 3? (a) 1978 (b) 1974 (c) 1976 (d) 1973
  6. 6.What is the smallest number you must add to 181 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
  7. 7.What is the smallest number you must add to 611 to make it divisible by 9? (a) 8 (b) 2 (c) 1 (d) 3
  8. 8.What is the smallest number you must add to 318 to make it divisible by 11? (a) 1 (b) 3 (c) 10 (d) 2
  9. 9.Which of these numbers is divisible by 11? (a) 6258 (b) 6266 (c) 6259 (d) 6279
  10. 10.What is the remainder when 222 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  11. 11.What is the smallest number you must add to 149 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  12. 12.Which of these numbers is divisible by 4? (a) 1589 (b) 1595 (c) 1599 (d) 1592
  13. 13.What is the units digit of 2^22? (a) 8 (b) 2 (c) 6 (d) 4
  14. 14.What is the smallest number you must add to 297 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
  15. 15.What is the units digit of 9^50? (a) 0 (b) 2 (c) 9 (d) 1
  16. 16.What is the remainder when 142 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
  17. 17.What is the smallest number you must add to 523 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
  18. 18.Which of these numbers is divisible by 3? (a) 1403 (b) 1409 (c) 1406 (d) 1404
  19. 19.What is the remainder when 203 is divided by 10? (a) 7 (b) 4 (c) 2 (d) 3
  20. 20.What is the units digit of 3^32? (a) 1 (b) 7 (c) 9 (d) 3
  21. 21.What is the smallest number you must add to 296 to make it divisible by 7? (a) 6 (b) 5 (c) 2 (d) 4
  22. 22.What is the units digit of 8^18? (a) 6 (b) 4 (c) 2 (d) 8
  23. 23.What is the remainder when 309 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  24. 24.Which of these numbers is divisible by 11? (a) 3100 (b) 3079 (c) 3089 (d) 3080
  25. 25.What is the units digit of 8^38? (a) 8 (b) 2 (c) 6 (d) 4
  26. 26.Which of these numbers is divisible by 9? (a) 6516 (b) 6515 (c) 6529 (d) 6524
  27. 27.Which of these numbers is divisible by 8? (a) 6489 (b) 6488 (c) 6497 (d) 6481
  28. 28.What is the units digit of 7^59? (a) 1 (b) 7 (c) 3 (d) 9
  29. 29.What is the smallest number you must add to 273 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  30. 30.What is the smallest number you must add to 169 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 4
  31. 31.Which of these numbers is divisible by 9? (a) 860 (b) 871 (c) 879 (d) 864
  32. 32.What is the remainder when 392 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
  33. 33.What is the remainder when 306 is divided by 11? (a) 8 (b) 2 (c) 10 (d) 9
  34. 34.What is the remainder when 289 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
  35. 35.What is the smallest number you must add to 197 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  36. 36.What is the remainder when 563 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 4
  37. 37.Which of these numbers is divisible by 11? (a) 8605 (b) 8626 (c) 8613 (d) 8623
  38. 38.Which of these numbers is divisible by 4? (a) 636 (b) 638 (c) 634 (d) 642
  39. 39.Which of these numbers is divisible by 11? (a) 9353 (b) 9350 (c) 9364 (d) 9347
  40. 40.What is the remainder when 322 is divided by 9? (a) 2 (b) 7 (c) 6 (d) 8

Answer key

  1. 3188
  2. 7
  3. 4
  4. 5
  5. 1974
  6. 5
  7. 1
  8. 1
  9. 6259
  10. 2
  11. 3
  12. 1592
  13. 4
  14. 7
  15. 1
  16. 2
  17. 5
  18. 1404
  19. 3
  20. 1
  21. 5
  22. 4
  23. 1
  24. 3080
  25. 4
  26. 6516
  27. 6488
  28. 3
  29. 3
  30. 5
  31. 864
  32. 2
  33. 9
  34. 9
  35. 3
  36. 5
  37. 8613
  38. 636
  39. 9350
  40. 7

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