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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 51 · 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 11? (a) 5903 (b) 5885 (c) 5887 (d) 5876
  2. 2.What is the smallest number you must add to 157 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  3. 3.Which of these numbers is divisible by 11? (a) 977 (b) 948 (c) 957 (d) 964
  4. 4.What is the remainder when 101 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5
  5. 5.What is the smallest number you must add to 652 to make it divisible by 11? (a) 3 (b) 9 (c) 7 (d) 8
  6. 6.What is the smallest number you must add to 822 to make it divisible by 11? (a) 2 (b) 4 (c) 3 (d) 8
  7. 7.What is the remainder when 209 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 1
  8. 8.What is the remainder when 489 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
  9. 9.What is the units digit of 2^27? (a) 2 (b) 4 (c) 6 (d) 8
  10. 10.What is the units digit of 8^22? (a) 2 (b) 8 (c) 4 (d) 6
  11. 11.What is the smallest number you must add to 688 to make it divisible by 10? (a) 1 (b) 3 (c) 2 (d) 8
  12. 12.What is the remainder when 181 is divided by 12? (a) 1 (b) 11 (c) 2 (d) 0
  13. 13.What is the smallest number you must add to 374 to make it divisible by 7? (a) 6 (b) 4 (c) 3 (d) 5
  14. 14.What is the remainder when 217 is divided by 10? (a) 6 (b) 8 (c) 7 (d) 3
  15. 15.What is the remainder when 469 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
  16. 16.What is the smallest number you must add to 227 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
  17. 17.What is the remainder when 41 is divided by 3? (a) 2 (b) 1 (c) 0
  18. 18.Which of these numbers is divisible by 3? (a) 618 (b) 619 (c) 617 (d) 622
  19. 19.What is the units digit of 8^11? (a) 8 (b) 4 (c) 2 (d) 6
  20. 20.What is the remainder when 350 is divided by 12? (a) 10 (b) 2 (c) 3 (d) 1
  21. 21.What is the remainder when 206 is divided by 9? (a) 6 (b) 7 (c) 1 (d) 8
  22. 22.What is the remainder when 471 is divided by 9? (a) 4 (b) 2 (c) 6 (d) 3
  23. 23.What is the remainder when 269 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
  24. 24.What is the smallest number you must add to 332 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  25. 25.Which of these numbers is divisible by 3? (a) 1780 (b) 1787 (c) 1782 (d) 1783
  26. 26.What is the smallest number you must add to 508 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 7
  27. 27.Which of these numbers is divisible by 8? (a) 2836 (b) 2841 (c) 2853 (d) 2840
  28. 28.What is the smallest number you must add to 202 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  29. 29.Which of these numbers is divisible by 11? (a) 2387 (b) 2391 (c) 2403 (d) 2382
  30. 30.What is the smallest number you must add to 233 to make it divisible by 3? (a) 1 (b) 2
  31. 31.What is the smallest number you must add to 787 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 4
  32. 32.What is the smallest number you must add to 111 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2
  33. 33.What is the units digit of 4^30? (a) 7 (b) 4 (c) 5 (d) 6
  34. 34.What is the remainder when 133 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
  35. 35.What is the smallest number you must add to 141 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
  36. 36.What is the remainder when 598 is divided by 11? (a) 5 (b) 7 (c) 3 (d) 4
  37. 37.What is the remainder when 83 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
  38. 38.What is the units digit of 2^51? (a) 4 (b) 8 (c) 6 (d) 2
  39. 39.Which of these numbers is divisible by 11? (a) 1059 (b) 1052 (c) 1077 (d) 1056
  40. 40.What is the units digit of 2^17? (a) 8 (b) 4 (c) 6 (d) 2

Answer key

  1. 5885
  2. 3
  3. 957
  4. 5
  5. 8
  6. 3
  7. 6
  8. 3
  9. 8
  10. 4
  11. 2
  12. 1
  13. 4
  14. 7
  15. 1
  16. 4
  17. 2
  18. 618
  19. 2
  20. 2
  21. 8
  22. 3
  23. 3
  24. 3
  25. 1782
  26. 5
  27. 2840
  28. 3
  29. 2387
  30. 1
  31. 5
  32. 1
  33. 6
  34. 5
  35. 4
  36. 4
  37. 3
  38. 8
  39. 1056
  40. 2

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