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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 333 · 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 59 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  2. 2.What is the units digit of 7^55? (a) 3 (b) 1 (c) 7 (d) 9
  3. 3.What is the smallest number you must add to 143 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  4. 4.What is the units digit of 7^11? (a) 7 (b) 3 (c) 1 (d) 9
  5. 5.What is the remainder when 441 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
  6. 6.Which of these numbers is divisible by 11? (a) 7714 (b) 7734 (c) 7724 (d) 7722
  7. 7.Which of these numbers is divisible by 3? (a) 1317 (b) 1316 (c) 1322 (d) 1318
  8. 8.Which of these numbers is divisible by 9? (a) 7272 (b) 7271 (c) 7279 (d) 7287
  9. 9.Which of these numbers is divisible by 6? (a) 762 (b) 760 (c) 771 (d) 767
  10. 10.Which of these numbers is divisible by 3? (a) 701 (b) 704 (c) 702 (d) 706
  11. 11.What is the units digit of 3^28? (a) 3 (b) 9 (c) 1 (d) 7
  12. 12.What is the smallest number you must add to 248 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  13. 13.What is the smallest number you must add to 509 to make it divisible by 7? (a) 5 (b) 1 (c) 3 (d) 2
  14. 14.Which of these numbers is divisible by 9? (a) 1158 (b) 1169 (c) 1147 (d) 1152
  15. 15.What is the units digit of 9^42? (a) 2 (b) 0 (c) 1 (d) 9
  16. 16.Which of these numbers is divisible by 3? (a) 1382 (b) 1377 (c) 1378 (d) 1375
  17. 17.Which of these numbers is divisible by 9? (a) 6797 (b) 6804 (c) 6819 (d) 6811
  18. 18.What is the remainder when 100 is divided by 3? (a) 0 (b) 2 (c) 1
  19. 19.What is the smallest number you must add to 486 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 2
  20. 20.What is the units digit of 8^16? (a) 2 (b) 4 (c) 6 (d) 8
  21. 21.What is the smallest number you must add to 724 to make it divisible by 12? (a) 7 (b) 4 (c) 8 (d) 9
  22. 22.Which of these numbers is divisible by 3? (a) 2345 (b) 2351 (c) 2348 (d) 2346
  23. 23.What is the smallest number you must add to 629 to make it divisible by 8? (a) 5 (b) 2 (c) 3 (d) 4
  24. 24.What is the units digit of 8^27? (a) 6 (b) 8 (c) 2 (d) 4
  25. 25.What is the smallest number you must add to 364 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  26. 26.Which of these numbers is divisible by 11? (a) 5676 (b) 5694 (c) 5671 (d) 5677
  27. 27.Which of these numbers is divisible by 6? (a) 2406 (b) 2407 (c) 2405 (d) 2417
  28. 28.What is the smallest number you must add to 427 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
  29. 29.What is the units digit of 9^20? (a) 9 (b) 2 (c) 0 (d) 1
  30. 30.What is the units digit of 3^29? (a) 7 (b) 9 (c) 1 (d) 3
  31. 31.Which of these numbers is divisible by 6? (a) 1387 (b) 1381 (c) 1394 (d) 1386
  32. 32.Which of these numbers is divisible by 6? (a) 339 (b) 342 (c) 351 (d) 347
  33. 33.What is the smallest number you must add to 248 to make it divisible by 12? (a) 8 (b) 3 (c) 5 (d) 4
  34. 34.Which of these numbers is divisible by 4? (a) 847 (b) 844 (c) 851 (d) 841
  35. 35.What is the smallest number you must add to 127 to make it divisible by 8? (a) 3 (b) 2 (c) 7 (d) 1
  36. 36.What is the remainder when 401 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
  37. 37.What is the smallest number you must add to 394 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
  38. 38.What is the smallest number you must add to 217 to make it divisible by 12? (a) 11 (b) 9 (c) 1 (d) 10
  39. 39.Which of these numbers is divisible by 9? (a) 506 (b) 514 (c) 504 (d) 500
  40. 40.What is the remainder when 445 is divided by 12? (a) 11 (b) 2 (c) 1 (d) 0

Answer key

  1. 3
  2. 3
  3. 1
  4. 3
  5. 3
  6. 7722
  7. 1317
  8. 7272
  9. 762
  10. 702
  11. 1
  12. 2
  13. 2
  14. 1152
  15. 1
  16. 1377
  17. 6804
  18. 1
  19. 9
  20. 6
  21. 8
  22. 2346
  23. 3
  24. 2
  25. 1
  26. 5676
  27. 2406
  28. 5
  29. 1
  30. 3
  31. 1386
  32. 342
  33. 4
  34. 844
  35. 1
  36. 5
  37. 2
  38. 11
  39. 504
  40. 1

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