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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 739 · 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 smallest number you must add to 391 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
  2. 2.What is the units digit of 3^19? (a) 1 (b) 3 (c) 7 (d) 9
  3. 3.What is the smallest number you must add to 403 to make it divisible by 10? (a) 6 (b) 8 (c) 3 (d) 7
  4. 4.What is the units digit of 4^49? (a) 5 (b) 3 (c) 4 (d) 6
  5. 5.What is the units digit of 2^56? (a) 4 (b) 8 (c) 2 (d) 6
  6. 6.What is the smallest number you must add to 125 to make it divisible by 8? (a) 2 (b) 5 (c) 3 (d) 4
  7. 7.Which of these numbers is divisible by 9? (a) 4077 (b) 4087 (c) 4085 (d) 4075
  8. 8.What is the units digit of 4^51? (a) 5 (b) 3 (c) 6 (d) 4
  9. 9.What is the smallest number you must add to 189 to make it divisible by 10? (a) 9 (b) 2 (c) 1 (d) 3
  10. 10.What is the remainder when 281 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
  11. 11.What is the remainder when 580 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 5
  12. 12.What is the smallest number you must add to 501 to make it divisible by 10? (a) 8 (b) 9 (c) 7 (d) 1
  13. 13.What is the units digit of 8^52? (a) 8 (b) 2 (c) 4 (d) 6
  14. 14.What is the remainder when 212 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 7
  15. 15.What is the remainder when 186 is divided by 9? (a) 3 (b) 7 (c) 5 (d) 6
  16. 16.What is the smallest number you must add to 123 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
  17. 17.What is the units digit of 3^53? (a) 7 (b) 9 (c) 1 (d) 3
  18. 18.What is the smallest number you must add to 191 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 1
  19. 19.What is the smallest number you must add to 723 to make it divisible by 12? (a) 3 (b) 9 (c) 10 (d) 8
  20. 20.What is the remainder when 135 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
  21. 21.What is the units digit of 8^16? (a) 8 (b) 6 (c) 4 (d) 2
  22. 22.What is the remainder when 349 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  23. 23.What is the units digit of 2^45? (a) 2 (b) 6 (c) 4 (d) 8
  24. 24.What is the smallest number you must add to 367 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  25. 25.What is the remainder when 219 is divided by 11? (a) 1 (b) 9 (c) 8 (d) 10
  26. 26.What is the remainder when 381 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 0
  27. 27.What is the smallest number you must add to 629 to make it divisible by 10? (a) 2 (b) 1 (c) 9 (d) 3
  28. 28.Which of these numbers is divisible by 8? (a) 4609 (b) 4605 (c) 4618 (d) 4608
  29. 29.Which of these numbers is divisible by 8? (a) 3768 (b) 3778 (c) 3775 (d) 3766
  30. 30.What is the smallest number you must add to 687 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 7
  31. 31.What is the units digit of 7^50? (a) 3 (b) 7 (c) 9 (d) 1
  32. 32.Which of these numbers is divisible by 6? (a) 3878 (b) 3882 (c) 3891 (d) 3883
  33. 33.Which of these numbers is divisible by 8? (a) 1142 (b) 1127 (c) 1128 (d) 1132
  34. 34.What is the remainder when 366 is divided by 8? (a) 6 (b) 2 (c) 7 (d) 5
  35. 35.Which of these numbers is divisible by 11? (a) 4147 (b) 4146 (c) 4159 (d) 4153
  36. 36.What is the smallest number you must add to 149 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  37. 37.What is the remainder when 444 is divided by 11? (a) 5 (b) 7 (c) 3 (d) 4
  38. 38.What is the units digit of 9^48? (a) 1 (b) 2 (c) 9 (d) 0
  39. 39.What is the units digit of 7^18? (a) 3 (b) 7 (c) 1 (d) 9
  40. 40.Which of these numbers is divisible by 11? (a) 2291 (b) 2316 (c) 2306 (d) 2299

Answer key

  1. 1
  2. 7
  3. 7
  4. 4
  5. 6
  6. 3
  7. 4077
  8. 4
  9. 1
  10. 1
  11. 4
  12. 9
  13. 6
  14. 5
  15. 6
  16. 3
  17. 3
  18. 1
  19. 9
  20. 3
  21. 6
  22. 4
  23. 2
  24. 3
  25. 10
  26. 1
  27. 1
  28. 4608
  29. 3768
  30. 6
  31. 9
  32. 3882
  33. 1128
  34. 6
  35. 4147
  36. 3
  37. 4
  38. 1
  39. 9
  40. 2299

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