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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 485 · 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 87 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
  2. 2.What is the units digit of 4^16? (a) 6 (b) 7 (c) 5 (d) 4
  3. 3.What is the remainder when 367 is divided by 10? (a) 3 (b) 6 (c) 7 (d) 8
  4. 4.What is the units digit of 9^13? (a) 7 (b) 1 (c) 8 (d) 9
  5. 5.What is the remainder when 835 is divided by 11? (a) 10 (b) 9 (c) 1 (d) 8
  6. 6.What is the units digit of 4^20? (a) 7 (b) 4 (c) 5 (d) 6
  7. 7.What is the units digit of 3^47? (a) 9 (b) 3 (c) 7 (d) 1
  8. 8.Which of these numbers is divisible by 3? (a) 534 (b) 532 (c) 539 (d) 536
  9. 9.What is the remainder when 617 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 4
  10. 10.What is the remainder when 574 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 3
  11. 11.What is the units digit of 4^32? (a) 7 (b) 5 (c) 4 (d) 6
  12. 12.What is the units digit of 9^18? (a) 1 (b) 9 (c) 2 (d) 0
  13. 13.What is the units digit of 9^46? (a) 9 (b) 2 (c) 0 (d) 1
  14. 14.What is the remainder when 205 is divided by 7? (a) 3 (b) 2 (c) 5 (d) 1
  15. 15.What is the smallest number you must add to 379 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
  16. 16.Which of these numbers is divisible by 3? (a) 1650 (b) 1648 (c) 1652 (d) 1655
  17. 17.What is the remainder when 227 is divided by 9? (a) 7 (b) 2 (c) 3 (d) 1
  18. 18.Which of these numbers is divisible by 9? (a) 4572 (b) 4569 (c) 4584 (d) 4576
  19. 19.What is the remainder when 214 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  20. 20.Which of these numbers is divisible by 6? (a) 1614 (b) 1611 (c) 1622 (d) 1615
  21. 21.What is the smallest number you must add to 138 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 5
  22. 22.What is the smallest number you must add to 452 to make it divisible by 12? (a) 3 (b) 8 (c) 5 (d) 4
  23. 23.Which of these numbers is divisible by 3? (a) 1147 (b) 1151 (c) 1149 (d) 1153
  24. 24.What is the remainder when 341 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5
  25. 25.What is the smallest number you must add to 255 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 6
  26. 26.Which of these numbers is divisible by 3? (a) 2234 (b) 2239 (c) 2235 (d) 2237
  27. 27.What is the smallest number you must add to 182 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
  28. 28.What is the smallest number you must add to 95 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
  29. 29.What is the units digit of 9^57? (a) 1 (b) 8 (c) 7 (d) 9
  30. 30.What is the remainder when 137 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  31. 31.What is the remainder when 413 is divided by 8? (a) 3 (b) 4 (c) 5 (d) 6
  32. 32.What is the remainder when 116 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
  33. 33.What is the smallest number you must add to 575 to make it divisible by 11? (a) 3 (b) 8 (c) 7 (d) 9
  34. 34.What is the smallest number you must add to 651 to make it divisible by 12? (a) 8 (b) 9 (c) 10 (d) 3
  35. 35.What is the smallest number you must add to 213 to make it divisible by 12? (a) 9 (b) 3 (c) 4 (d) 2
  36. 36.What is the remainder when 299 is divided by 12? (a) 1 (b) 10 (c) 11 (d) 9
  37. 37.What is the units digit of 8^39? (a) 4 (b) 2 (c) 8 (d) 6
  38. 38.What is the remainder when 305 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
  39. 39.What is the units digit of 9^17? (a) 1 (b) 7 (c) 8 (d) 9
  40. 40.What is the remainder when 512 is divided by 7? (a) 6 (b) 1 (c) 0 (d) 2

Answer key

  1. 3
  2. 6
  3. 7
  4. 9
  5. 10
  6. 6
  7. 7
  8. 534
  9. 5
  10. 4
  11. 6
  12. 1
  13. 1
  14. 2
  15. 1
  16. 1650
  17. 2
  18. 4572
  19. 4
  20. 1614
  21. 2
  22. 4
  23. 1149
  24. 5
  25. 4
  26. 2235
  27. 4
  28. 1
  29. 9
  30. 1
  31. 5
  32. 1
  33. 8
  34. 9
  35. 3
  36. 11
  37. 2
  38. 5
  39. 9
  40. 1

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