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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 174 · 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 9? (a) 872 (b) 847 (c) 855 (d) 856
  2. 2.What is the units digit of 4^26? (a) 5 (b) 6 (c) 7 (d) 4
  3. 3.What is the smallest number you must add to 610 to make it divisible by 9? (a) 7 (b) 2 (c) 1 (d) 3
  4. 4.What is the units digit of 9^19? (a) 9 (b) 1 (c) 7 (d) 8
  5. 5.What is the units digit of 3^24? (a) 7 (b) 9 (c) 1 (d) 3
  6. 6.What is the remainder when 305 is divided by 7? (a) 5 (b) 3 (c) 6 (d) 4
  7. 7.Which of these numbers is divisible by 6? (a) 582 (b) 583 (c) 592 (d) 579
  8. 8.What is the units digit of 8^43? (a) 2 (b) 6 (c) 8 (d) 4
  9. 9.What is the remainder when 467 is divided by 11? (a) 4 (b) 5 (c) 6 (d) 7
  10. 10.What is the smallest number you must add to 211 to make it divisible by 3? (a) 1 (b) 2
  11. 11.What is the smallest number you must add to 148 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 4
  12. 12.Which of these numbers is divisible by 9? (a) 3600 (b) 3593 (c) 3612 (d) 3605
  13. 13.Which of these numbers is divisible by 8? (a) 2859 (b) 2849 (c) 2866 (d) 2856
  14. 14.What is the smallest number you must add to 163 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
  15. 15.What is the smallest number you must add to 53 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  16. 16.Which of these numbers is divisible by 6? (a) 1093 (b) 1081 (c) 1086 (d) 1087
  17. 17.Which of these numbers is divisible by 3? (a) 1660 (b) 1663 (c) 1659 (d) 1657
  18. 18.What is the units digit of 7^12? (a) 7 (b) 3 (c) 9 (d) 1
  19. 19.What is the remainder when 229 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  20. 20.What is the remainder when 850 is divided by 12? (a) 2 (b) 9 (c) 11 (d) 10
  21. 21.What is the remainder when 102 is divided by 10? (a) 1 (b) 8 (c) 2 (d) 3
  22. 22.Which of these numbers is divisible by 9? (a) 1088 (b) 1071 (c) 1077 (d) 1063
  23. 23.What is the smallest number you must add to 492 to make it divisible by 7? (a) 6 (b) 2 (c) 5 (d) 4
  24. 24.What is the remainder when 152 is divided by 10? (a) 2 (b) 3 (c) 1 (d) 8
  25. 25.What is the smallest number you must add to 235 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
  26. 26.Which of these numbers is divisible by 4? (a) 2201 (b) 2198 (c) 2206 (d) 2200
  27. 27.What is the smallest number you must add to 139 to make it divisible by 3? (a) 2 (b) 1
  28. 28.What is the smallest number you must add to 132 to make it divisible by 7? (a) 1 (b) 6 (c) 3 (d) 2
  29. 29.Which of these numbers is divisible by 3? (a) 1117 (b) 1115 (c) 1113 (d) 1111
  30. 30.What is the smallest number you must add to 58 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  31. 31.What is the remainder when 486 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
  32. 32.What is the smallest number you must add to 270 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  33. 33.What is the remainder when 476 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1
  34. 34.What is the remainder when 284 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
  35. 35.What is the remainder when 469 is divided by 11? (a) 6 (b) 8 (c) 4 (d) 7
  36. 36.What is the units digit of 9^48? (a) 2 (b) 9 (c) 1 (d) 0
  37. 37.What is the smallest number you must add to 49 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  38. 38.What is the remainder when 904 is divided by 12? (a) 5 (b) 4 (c) 3 (d) 8
  39. 39.What is the smallest number you must add to 305 to make it divisible by 6? (a) 2 (b) 5 (c) 1 (d) 3
  40. 40.What is the smallest number you must add to 352 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1

Answer key

  1. 855
  2. 6
  3. 2
  4. 9
  5. 1
  6. 4
  7. 582
  8. 2
  9. 5
  10. 2
  11. 2
  12. 3600
  13. 2856
  14. 5
  15. 2
  16. 1086
  17. 1659
  18. 1
  19. 4
  20. 10
  21. 2
  22. 1071
  23. 5
  24. 2
  25. 5
  26. 2200
  27. 2
  28. 1
  29. 1113
  30. 2
  31. 6
  32. 2
  33. 8
  34. 4
  35. 7
  36. 1
  37. 3
  38. 4
  39. 1
  40. 3

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