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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 377 · 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 units digit of 3^33? (a) 7 (b) 1 (c) 3 (d) 9
  2. 2.What is the smallest number you must add to 213 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  3. 3.What is the smallest number you must add to 163 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 1
  4. 4.What is the smallest number you must add to 546 to make it divisible by 12? (a) 7 (b) 8 (c) 5 (d) 6
  5. 5.Which of these numbers is divisible by 11? (a) 851 (b) 844 (c) 847 (d) 868
  6. 6.Which of these numbers is divisible by 9? (a) 7052 (b) 7040 (c) 7059 (d) 7047
  7. 7.Which of these numbers is divisible by 9? (a) 5112 (b) 5124 (c) 5119 (d) 5105
  8. 8.What is the smallest number you must add to 125 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  9. 9.Which of these numbers is divisible by 8? (a) 4929 (b) 4924 (c) 4919 (d) 4920
  10. 10.What is the remainder when 461 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
  11. 11.What is the units digit of 4^34? (a) 6 (b) 7 (c) 5 (d) 4
  12. 12.What is the remainder when 81 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  13. 13.What is the remainder when 183 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2
  14. 14.What is the units digit of 3^17? (a) 9 (b) 1 (c) 3 (d) 7
  15. 15.What is the smallest number you must add to 333 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
  16. 16.What is the smallest number you must add to 41 to make it divisible by 3? (a) 2 (b) 1
  17. 17.What is the smallest number you must add to 234 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  18. 18.What is the remainder when 633 is divided by 8? (a) 2 (b) 1 (c) 7 (d) 0
  19. 19.What is the remainder when 393 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 7
  20. 20.What is the units digit of 7^45? (a) 3 (b) 1 (c) 9 (d) 7
  21. 21.What is the smallest number you must add to 146 to make it divisible by 3? (a) 2 (b) 1
  22. 22.What is the units digit of 7^50? (a) 1 (b) 3 (c) 9 (d) 7
  23. 23.What is the smallest number you must add to 452 to make it divisible by 11? (a) 1 (b) 10 (c) 9 (d) 8
  24. 24.Which of these numbers is divisible by 9? (a) 972 (b) 986 (c) 973 (d) 971
  25. 25.Which of these numbers is divisible by 8? (a) 5479 (b) 5457 (c) 5465 (d) 5464
  26. 26.Which of these numbers is divisible by 8? (a) 1091 (b) 1088 (c) 1103 (d) 1082
  27. 27.What is the units digit of 7^49? (a) 1 (b) 7 (c) 9 (d) 3
  28. 28.Which of these numbers is divisible by 4? (a) 738 (b) 741 (c) 733 (d) 736
  29. 29.What is the smallest number you must add to 177 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 2
  30. 30.What is the smallest number you must add to 571 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 7
  31. 31.What is the remainder when 362 is divided by 12? (a) 3 (b) 1 (c) 10 (d) 2
  32. 32.Which of these numbers is divisible by 11? (a) 2695 (b) 2702 (c) 2686 (d) 2710
  33. 33.What is the smallest number you must add to 126 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  34. 34.Which of these numbers is divisible by 4? (a) 555 (b) 546 (c) 548 (d) 549
  35. 35.What is the smallest number you must add to 618 to make it divisible by 11? (a) 2 (b) 9 (c) 8 (d) 10
  36. 36.What is the units digit of 2^41? (a) 4 (b) 2 (c) 8 (d) 6
  37. 37.What is the smallest number you must add to 297 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  38. 38.Which of these numbers is divisible by 4? (a) 222 (b) 227 (c) 224 (d) 230
  39. 39.What is the smallest number you must add to 293 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  40. 40.Which of these numbers is divisible by 9? (a) 1124 (b) 1139 (c) 1125 (d) 1133

Answer key

  1. 3
  2. 2
  3. 5
  4. 6
  5. 847
  6. 7047
  7. 5112
  8. 3
  9. 4920
  10. 5
  11. 6
  12. 1
  13. 1
  14. 3
  15. 3
  16. 1
  17. 1
  18. 1
  19. 6
  20. 7
  21. 1
  22. 9
  23. 10
  24. 972
  25. 5464
  26. 1088
  27. 7
  28. 736
  29. 3
  30. 5
  31. 2
  32. 2695
  33. 4
  34. 548
  35. 9
  36. 2
  37. 3
  38. 224
  39. 2
  40. 1125

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