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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 186 · 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 382 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 2
  2. 2.Which of these numbers is divisible by 3? (a) 1746 (b) 1745 (c) 1750 (d) 1747
  3. 3.What is the smallest number you must add to 418 to make it divisible by 12? (a) 3 (b) 1 (c) 2 (d) 10
  4. 4.What is the units digit of 8^32? (a) 6 (b) 4 (c) 8 (d) 2
  5. 5.What is the remainder when 349 is divided by 11? (a) 7 (b) 8 (c) 9 (d) 3
  6. 6.What is the remainder when 274 is divided by 11? (a) 9 (b) 1 (c) 10 (d) 8
  7. 7.What is the units digit of 8^50? (a) 4 (b) 8 (c) 6 (d) 2
  8. 8.What is the remainder when 339 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
  9. 9.Which of these numbers is divisible by 4? (a) 1618 (b) 1611 (c) 1615 (d) 1612
  10. 10.What is the units digit of 2^38? (a) 6 (b) 8 (c) 4 (d) 2
  11. 11.What is the smallest number you must add to 258 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  12. 12.What is the smallest number you must add to 282 to make it divisible by 8? (a) 2 (b) 6 (c) 5 (d) 7
  13. 13.What is the smallest number you must add to 179 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
  14. 14.Which of these numbers is divisible by 4? (a) 293 (b) 289 (c) 287 (d) 288
  15. 15.What is the remainder when 114 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 1
  16. 16.What is the remainder when 102 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
  17. 17.What is the smallest number you must add to 229 to make it divisible by 3? (a) 2 (b) 1
  18. 18.What is the remainder when 142 is divided by 7? (a) 5 (b) 3 (c) 1 (d) 2
  19. 19.What is the units digit of 9^12? (a) 9 (b) 0 (c) 1 (d) 2
  20. 20.What is the smallest number you must add to 193 to make it divisible by 3? (a) 1 (b) 2
  21. 21.Which of these numbers is divisible by 3? (a) 2074 (b) 2070 (c) 2071 (d) 2069
  22. 22.What is the units digit of 2^44? (a) 8 (b) 4 (c) 2 (d) 6
  23. 23.What is the smallest number you must add to 118 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  24. 24.What is the units digit of 9^27? (a) 7 (b) 1 (c) 9 (d) 8
  25. 25.What is the remainder when 452 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2
  26. 26.Which of these numbers is divisible by 6? (a) 4719 (b) 4725 (c) 4712 (d) 4716
  27. 27.What is the remainder when 201 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
  28. 28.What is the units digit of 8^53? (a) 4 (b) 6 (c) 8 (d) 2
  29. 29.Which of these numbers is divisible by 9? (a) 3883 (b) 3866 (c) 3870 (d) 3873
  30. 30.What is the smallest number you must add to 137 to make it divisible by 3? (a) 2 (b) 1
  31. 31.Which of these numbers is divisible by 6? (a) 1200 (b) 1207 (c) 1195 (d) 1202
  32. 32.What is the remainder when 141 is divided by 8? (a) 4 (b) 6 (c) 3 (d) 5
  33. 33.Which of these numbers is divisible by 9? (a) 2707 (b) 2709 (c) 2714 (d) 2724
  34. 34.What is the units digit of 3^47? (a) 7 (b) 3 (c) 1 (d) 9
  35. 35.What is the smallest number you must add to 462 to make it divisible by 12? (a) 7 (b) 8 (c) 5 (d) 6
  36. 36.What is the remainder when 183 is divided by 12? (a) 9 (b) 4 (c) 3 (d) 2
  37. 37.Which of these numbers is divisible by 11? (a) 1993 (b) 1989 (c) 1980 (d) 1979
  38. 38.Which of these numbers is divisible by 4? (a) 1925 (b) 1920 (c) 1917 (d) 1923
  39. 39.What is the remainder when 233 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
  40. 40.What is the remainder when 218 is divided by 3? (a) 1 (b) 0 (c) 2

Answer key

  1. 2
  2. 1746
  3. 2
  4. 6
  5. 8
  6. 10
  7. 4
  8. 3
  9. 1612
  10. 4
  11. 2
  12. 6
  13. 1
  14. 288
  15. 2
  16. 2
  17. 2
  18. 2
  19. 1
  20. 2
  21. 2070
  22. 6
  23. 2
  24. 9
  25. 2
  26. 4716
  27. 1
  28. 8
  29. 3870
  30. 1
  31. 1200
  32. 5
  33. 2709
  34. 7
  35. 6
  36. 3
  37. 1980
  38. 1920
  39. 1
  40. 2

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