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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 279 · 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 757 is divided by 11? (a) 2 (b) 9 (c) 8 (d) 10
  2. 2.What is the units digit of 8^36? (a) 8 (b) 4 (c) 2 (d) 6
  3. 3.Which of these numbers is divisible by 11? (a) 5721 (b) 5733 (c) 5745 (d) 5731
  4. 4.What is the remainder when 230 is divided by 8? (a) 5 (b) 6 (c) 2 (d) 7
  5. 5.Which of these numbers is divisible by 4? (a) 491 (b) 487 (c) 493 (d) 488
  6. 6.What is the units digit of 2^41? (a) 6 (b) 2 (c) 8 (d) 4
  7. 7.What is the smallest number you must add to 653 to make it divisible by 9? (a) 5 (b) 3 (c) 4 (d) 6
  8. 8.What is the smallest number you must add to 157 to make it divisible by 8? (a) 3 (b) 2 (c) 5 (d) 4
  9. 9.Which of these numbers is divisible by 6? (a) 1761 (b) 1769 (c) 1775 (d) 1764
  10. 10.What is the smallest number you must add to 215 to make it divisible by 9? (a) 8 (b) 1 (c) 2 (d) 3
  11. 11.What is the remainder when 86 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
  12. 12.What is the units digit of 4^45? (a) 3 (b) 5 (c) 4 (d) 6
  13. 13.What is the units digit of 2^22? (a) 2 (b) 8 (c) 6 (d) 4
  14. 14.Which of these numbers is divisible by 4? (a) 3075 (b) 3077 (c) 3072 (d) 3069
  15. 15.What is the units digit of 8^27? (a) 4 (b) 2 (c) 8 (d) 6
  16. 16.What is the smallest number you must add to 589 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 2
  17. 17.What is the smallest number you must add to 215 to make it divisible by 3? (a) 2 (b) 1
  18. 18.What is the units digit of 9^58? (a) 1 (b) 0 (c) 9 (d) 2
  19. 19.What is the remainder when 655 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 5
  20. 20.Which of these numbers is divisible by 6? (a) 4083 (b) 4072 (c) 4077 (d) 4074
  21. 21.Which of these numbers is divisible by 9? (a) 3537 (b) 3542 (c) 3547 (d) 3535
  22. 22.What is the remainder when 344 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
  23. 23.Which of these numbers is divisible by 9? (a) 401 (b) 395 (c) 412 (d) 396
  24. 24.What is the remainder when 662 is divided by 9? (a) 7 (b) 4 (c) 6 (d) 5
  25. 25.What is the smallest number you must add to 641 to make it divisible by 10? (a) 1 (b) 8 (c) 7 (d) 9
  26. 26.Which of these numbers is divisible by 8? (a) 2318 (b) 2331 (c) 2325 (d) 2320
  27. 27.What is the smallest number you must add to 232 to make it divisible by 12? (a) 7 (b) 4 (c) 8 (d) 9
  28. 28.What is the remainder when 212 is divided by 3? (a) 1 (b) 0 (c) 2
  29. 29.Which of these numbers is divisible by 4? (a) 1799 (b) 1803 (c) 1796 (d) 1795
  30. 30.What is the smallest number you must add to 950 to make it divisible by 12? (a) 11 (b) 2 (c) 9 (d) 10
  31. 31.What is the units digit of 3^10? (a) 7 (b) 9 (c) 1 (d) 3
  32. 32.What is the smallest number you must add to 615 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 7
  33. 33.What is the smallest number you must add to 182 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  34. 34.Which of these numbers is divisible by 8? (a) 980 (b) 986 (c) 984 (d) 993
  35. 35.What is the smallest number you must add to 661 to make it divisible by 11? (a) 1 (b) 10 (c) 8 (d) 9
  36. 36.What is the smallest number you must add to 857 to make it divisible by 11? (a) 1 (b) 10 (c) 2 (d) 3
  37. 37.What is the smallest number you must add to 392 to make it divisible by 12? (a) 8 (b) 5 (c) 3 (d) 4
  38. 38.Which of these numbers is divisible by 11? (a) 4866 (b) 4851 (c) 4861 (d) 4847
  39. 39.What is the units digit of 8^23? (a) 8 (b) 2 (c) 4 (d) 6
  40. 40.Which of these numbers is divisible by 11? (a) 633 (b) 654 (c) 642 (d) 638

Answer key

  1. 9
  2. 6
  3. 5731
  4. 6
  5. 488
  6. 2
  7. 4
  8. 3
  9. 1764
  10. 1
  11. 2
  12. 4
  13. 4
  14. 3072
  15. 2
  16. 3
  17. 1
  18. 1
  19. 6
  20. 4074
  21. 3537
  22. 4
  23. 396
  24. 5
  25. 9
  26. 2320
  27. 8
  28. 2
  29. 1796
  30. 10
  31. 9
  32. 1
  33. 3
  34. 984
  35. 10
  36. 1
  37. 4
  38. 4851
  39. 2
  40. 638

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