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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 371 · 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) 313 (b) 310 (c) 318 (d) 312
  2. 2.What is the smallest number you must add to 475 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
  3. 3.What is the smallest number you must add to 201 to make it divisible by 8? (a) 7 (b) 5 (c) 1 (d) 6
  4. 4.Which of these numbers is divisible by 3? (a) 370 (b) 373 (c) 369 (d) 367
  5. 5.What is the smallest number you must add to 326 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
  6. 6.What is the units digit of 8^55? (a) 4 (b) 2 (c) 6 (d) 8
  7. 7.What is the units digit of 3^16? (a) 1 (b) 3 (c) 9 (d) 7
  8. 8.Which of these numbers is divisible by 11? (a) 8863 (b) 8867 (c) 8855 (d) 8853
  9. 9.What is the units digit of 7^19? (a) 7 (b) 3 (c) 9 (d) 1
  10. 10.What is the remainder when 134 is divided by 7? (a) 1 (b) 6 (c) 2 (d) 0
  11. 11.What is the units digit of 2^24? (a) 4 (b) 8 (c) 6 (d) 2
  12. 12.What is the remainder when 482 is divided by 10? (a) 3 (b) 8 (c) 2 (d) 1
  13. 13.What is the smallest number you must add to 181 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  14. 14.What is the smallest number you must add to 422 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 2
  15. 15.Which of these numbers is divisible by 4? (a) 901 (b) 900 (c) 898 (d) 906
  16. 16.What is the remainder when 458 is divided by 12? (a) 10 (b) 3 (c) 2 (d) 1
  17. 17.What is the units digit of 2^20? (a) 2 (b) 6 (c) 4 (d) 8
  18. 18.What is the smallest number you must add to 100 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2
  19. 19.What is the remainder when 698 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
  20. 20.Which of these numbers is divisible by 8? (a) 2308 (b) 2298 (c) 2313 (d) 2304
  21. 21.What is the units digit of 4^13? (a) 5 (b) 3 (c) 6 (d) 4
  22. 22.What is the remainder when 401 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
  23. 23.Which of these numbers is divisible by 6? (a) 1456 (b) 1458 (c) 1466 (d) 1461
  24. 24.What is the remainder when 887 is divided by 11? (a) 4 (b) 7 (c) 6 (d) 8
  25. 25.Which of these numbers is divisible by 3? (a) 2605 (b) 2599 (c) 2601 (d) 2603
  26. 26.What is the remainder when 142 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
  27. 27.What is the remainder when 110 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
  28. 28.Which of these numbers is divisible by 4? (a) 662 (b) 660 (c) 667 (d) 659
  29. 29.What is the smallest number you must add to 208 to make it divisible by 3? (a) 2 (b) 1
  30. 30.What is the smallest number you must add to 372 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9
  31. 31.What is the remainder when 399 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 2
  32. 32.What is the remainder when 847 is divided by 12? (a) 5 (b) 8 (c) 6 (d) 7
  33. 33.What is the smallest number you must add to 277 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 3
  34. 34.What is the smallest number you must add to 372 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  35. 35.Which of these numbers is divisible by 8? (a) 1665 (b) 1674 (c) 1661 (d) 1664
  36. 36.What is the smallest number you must add to 267 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
  37. 37.What is the remainder when 532 is divided by 11? (a) 7 (b) 5 (c) 3 (d) 4
  38. 38.What is the remainder when 150 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
  39. 39.What is the remainder when 107 is divided by 9? (a) 8 (b) 6 (c) 7 (d) 1
  40. 40.What is the remainder when 141 is divided by 8? (a) 5 (b) 4 (c) 6 (d) 3

Answer key

  1. 312
  2. 5
  3. 7
  4. 369
  5. 4
  6. 2
  7. 1
  8. 8855
  9. 3
  10. 1
  11. 6
  12. 2
  13. 4
  14. 10
  15. 900
  16. 2
  17. 6
  18. 5
  19. 8
  20. 2304
  21. 4
  22. 1
  23. 1458
  24. 7
  25. 2601
  26. 2
  27. 2
  28. 660
  29. 2
  30. 8
  31. 3
  32. 7
  33. 3
  34. 3
  35. 1664
  36. 8
  37. 4
  38. 6
  39. 8
  40. 5

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