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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 659 · 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 163 to make it divisible by 3? (a) 1 (b) 2
  2. 2.What is the units digit of 8^47? (a) 8 (b) 2 (c) 4 (d) 6
  3. 3.What is the remainder when 218 is divided by 3? (a) 2 (b) 0 (c) 1
  4. 4.What is the smallest number you must add to 237 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
  5. 5.What is the units digit of 9^10? (a) 9 (b) 2 (c) 0 (d) 1
  6. 6.What is the remainder when 650 is divided by 11? (a) 2 (b) 10 (c) 1 (d) 0
  7. 7.What is the units digit of 2^29? (a) 4 (b) 2 (c) 8 (d) 6
  8. 8.What is the remainder when 235 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  9. 9.What is the smallest number you must add to 131 to make it divisible by 3? (a) 2 (b) 1
  10. 10.What is the smallest number you must add to 137 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  11. 11.What is the units digit of 2^22? (a) 4 (b) 2 (c) 6 (d) 8
  12. 12.What is the remainder when 202 is divided by 7? (a) 1 (b) 6 (c) 5 (d) 4
  13. 13.What is the units digit of 8^37? (a) 2 (b) 6 (c) 8 (d) 4
  14. 14.What is the units digit of 2^36? (a) 6 (b) 8 (c) 2 (d) 4
  15. 15.What is the remainder when 431 is divided by 11? (a) 2 (b) 1 (c) 3 (d) 9
  16. 16.What is the smallest number you must add to 458 to make it divisible by 7? (a) 3 (b) 6 (c) 4 (d) 5
  17. 17.What is the smallest number you must add to 372 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
  18. 18.What is the units digit of 7^58? (a) 7 (b) 1 (c) 3 (d) 9
  19. 19.What is the smallest number you must add to 422 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
  20. 20.What is the remainder when 327 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
  21. 21.What is the remainder when 160 is divided by 7? (a) 4 (b) 1 (c) 6 (d) 5
  22. 22.What is the units digit of 7^60? (a) 7 (b) 9 (c) 3 (d) 1
  23. 23.What is the units digit of 4^34? (a) 4 (b) 6 (c) 7 (d) 5
  24. 24.What is the units digit of 3^23? (a) 1 (b) 7 (c) 9 (d) 3
  25. 25.What is the smallest number you must add to 548 to make it divisible by 10? (a) 1 (b) 8 (c) 2 (d) 3
  26. 26.What is the remainder when 241 is divided by 3? (a) 2 (b) 0 (c) 1
  27. 27.What is the units digit of 9^13? (a) 9 (b) 1 (c) 7 (d) 8
  28. 28.What is the smallest number you must add to 487 to make it divisible by 7? (a) 2 (b) 3 (c) 5 (d) 4
  29. 29.Which of these numbers is divisible by 8? (a) 1814 (b) 1800 (c) 1795 (d) 1802
  30. 30.Which of these numbers is divisible by 9? (a) 2006 (b) 2021 (c) 2008 (d) 2007
  31. 31.What is the remainder when 788 is divided by 10? (a) 8 (b) 7 (c) 2 (d) 9
  32. 32.What is the smallest number you must add to 173 to make it divisible by 10? (a) 8 (b) 6 (c) 7 (d) 3
  33. 33.What is the remainder when 71 is divided by 3? (a) 0 (b) 2 (c) 1
  34. 34.What is the smallest number you must add to 239 to make it divisible by 11? (a) 8 (b) 4 (c) 2 (d) 3
  35. 35.What is the remainder when 161 is divided by 3? (a) 1 (b) 2 (c) 0
  36. 36.What is the remainder when 292 is divided by 12? (a) 4 (b) 5 (c) 3 (d) 8
  37. 37.What is the remainder when 862 is divided by 11? (a) 3 (b) 4 (c) 5 (d) 7
  38. 38.What is the units digit of 9^50? (a) 9 (b) 0 (c) 1 (d) 2
  39. 39.What is the remainder when 577 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 4
  40. 40.What is the remainder when 234 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2

Answer key

  1. 2
  2. 2
  3. 2
  4. 3
  5. 1
  6. 1
  7. 2
  8. 3
  9. 1
  10. 3
  11. 4
  12. 6
  13. 8
  14. 6
  15. 2
  16. 4
  17. 3
  18. 9
  19. 4
  20. 3
  21. 6
  22. 1
  23. 6
  24. 7
  25. 2
  26. 1
  27. 9
  28. 3
  29. 1800
  30. 2007
  31. 8
  32. 7
  33. 2
  34. 3
  35. 2
  36. 4
  37. 4
  38. 1
  39. 5
  40. 2

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