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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 190 · 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) 832 (b) 831 (c) 835 (d) 838
  2. 2.What is the smallest number you must add to 77 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
  3. 3.What is the units digit of 8^15? (a) 2 (b) 6 (c) 8 (d) 4
  4. 4.What is the units digit of 4^37? (a) 5 (b) 3 (c) 4 (d) 6
  5. 5.What is the smallest number you must add to 477 to make it divisible by 10? (a) 7 (b) 3 (c) 2 (d) 4
  6. 6.What is the smallest number you must add to 134 to make it divisible by 3? (a) 2 (b) 1
  7. 7.What is the smallest number you must add to 131 to make it divisible by 3? (a) 1 (b) 2
  8. 8.What is the units digit of 3^20? (a) 9 (b) 3 (c) 1 (d) 7
  9. 9.What is the smallest number you must add to 515 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
  10. 10.What is the smallest number you must add to 418 to make it divisible by 7? (a) 1 (b) 3 (c) 5 (d) 2
  11. 11.What is the remainder when 57 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
  12. 12.What is the remainder when 125 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
  13. 13.What is the units digit of 3^51? (a) 7 (b) 9 (c) 1 (d) 3
  14. 14.What is the units digit of 7^24? (a) 3 (b) 9 (c) 7 (d) 1
  15. 15.What is the smallest number you must add to 218 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  16. 16.What is the smallest number you must add to 568 to make it divisible by 12? (a) 9 (b) 8 (c) 4 (d) 7
  17. 17.What is the units digit of 8^26? (a) 4 (b) 6 (c) 2 (d) 8
  18. 18.Which of these numbers is divisible by 9? (a) 1013 (b) 1027 (c) 1017 (d) 1023
  19. 19.What is the smallest number you must add to 377 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2
  20. 20.What is the units digit of 2^52? (a) 2 (b) 4 (c) 8 (d) 6
  21. 21.What is the units digit of 4^44? (a) 5 (b) 4 (c) 7 (d) 6
  22. 22.What is the units digit of 2^28? (a) 2 (b) 4 (c) 8 (d) 6
  23. 23.What is the smallest number you must add to 139 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  24. 24.What is the smallest number you must add to 126 to make it divisible by 10? (a) 3 (b) 6 (c) 5 (d) 4
  25. 25.Which of these numbers is divisible by 3? (a) 1046 (b) 1049 (c) 1052 (d) 1047
  26. 26.What is the units digit of 2^25? (a) 2 (b) 4 (c) 6 (d) 8
  27. 27.Which of these numbers is divisible by 4? (a) 316 (b) 321 (c) 313 (d) 319
  28. 28.What is the remainder when 278 is divided by 12? (a) 10 (b) 1 (c) 3 (d) 2
  29. 29.What is the smallest number you must add to 439 to make it divisible by 8? (a) 2 (b) 7 (c) 3 (d) 1
  30. 30.Which of these numbers is divisible by 6? (a) 1744 (b) 1746 (c) 1750 (d) 1757
  31. 31.What is the units digit of 8^54? (a) 8 (b) 2 (c) 6 (d) 4
  32. 32.What is the smallest number you must add to 461 to make it divisible by 8? (a) 2 (b) 3 (c) 5 (d) 4
  33. 33.What is the smallest number you must add to 76 to make it divisible by 7? (a) 2 (b) 6 (c) 3 (d) 1
  34. 34.Which of these numbers is divisible by 6? (a) 5284 (b) 5276 (c) 5274 (d) 5273
  35. 35.Which of these numbers is divisible by 8? (a) 745 (b) 759 (c) 744 (d) 738
  36. 36.Which of these numbers is divisible by 8? (a) 5400 (b) 5405 (c) 5415 (d) 5395
  37. 37.Which of these numbers is divisible by 4? (a) 1467 (b) 1459 (c) 1463 (d) 1460
  38. 38.What is the remainder when 101 is divided by 3? (a) 2 (b) 0 (c) 1
  39. 39.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
  40. 40.What is the smallest number you must add to 241 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 5

Answer key

  1. 832
  2. 1
  3. 2
  4. 4
  5. 3
  6. 1
  7. 1
  8. 1
  9. 5
  10. 2
  11. 2
  12. 5
  13. 7
  14. 1
  15. 2
  16. 8
  17. 4
  18. 1017
  19. 1
  20. 6
  21. 6
  22. 6
  23. 1
  24. 4
  25. 1047
  26. 2
  27. 316
  28. 2
  29. 1
  30. 1746
  31. 4
  32. 3
  33. 1
  34. 5274
  35. 744
  36. 5400
  37. 1460
  38. 2
  39. 2
  40. 5

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