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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 558 · 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 3? (a) 1452 (b) 1457 (c) 1450 (d) 1453
  2. 2.What is the smallest number you must add to 258 to make it divisible by 9? (a) 2 (b) 3 (c) 4 (d) 6
  3. 3.What is the units digit of 8^21? (a) 6 (b) 8 (c) 2 (d) 4
  4. 4.What is the units digit of 9^10? (a) 2 (b) 9 (c) 0 (d) 1
  5. 5.Which of these numbers is divisible by 6? (a) 1794 (b) 1805 (c) 1795 (d) 1793
  6. 6.What is the units digit of 8^50? (a) 2 (b) 8 (c) 6 (d) 4
  7. 7.Which of these numbers is divisible by 3? (a) 1540 (b) 1534 (c) 1536 (d) 1537
  8. 8.What is the remainder when 323 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
  9. 9.What is the smallest number you must add to 455 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
  10. 10.What is the smallest number you must add to 382 to make it divisible by 8? (a) 1 (b) 3 (c) 6 (d) 2
  11. 11.What is the units digit of 9^60? (a) 9 (b) 2 (c) 1 (d) 0
  12. 12.What is the smallest number you must add to 127 to make it divisible by 7? (a) 6 (b) 1 (c) 5 (d) 4
  13. 13.What is the smallest number you must add to 309 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 2
  14. 14.What is the units digit of 8^34? (a) 6 (b) 8 (c) 2 (d) 4
  15. 15.What is the smallest number you must add to 477 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
  16. 16.What is the smallest number you must add to 407 to make it divisible by 12? (a) 2 (b) 1 (c) 11 (d) 3
  17. 17.What is the remainder when 286 is divided by 9? (a) 2 (b) 8 (c) 6 (d) 7
  18. 18.What is the smallest number you must add to 107 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
  19. 19.What is the units digit of 8^25? (a) 4 (b) 8 (c) 6 (d) 2
  20. 20.What is the remainder when 223 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  21. 21.Which of these numbers is divisible by 4? (a) 537 (b) 542 (c) 540 (d) 545
  22. 22.What is the smallest number you must add to 186 to make it divisible by 7? (a) 2 (b) 5 (c) 4 (d) 3
  23. 23.What is the remainder when 127 is divided by 8? (a) 1 (b) 7 (c) 5 (d) 6
  24. 24.What is the smallest number you must add to 635 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 4
  25. 25.What is the remainder when 273 is divided by 9? (a) 4 (b) 6 (c) 2 (d) 3
  26. 26.What is the remainder when 108 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
  27. 27.What is the smallest number you must add to 308 to make it divisible by 10? (a) 2 (b) 1 (c) 8 (d) 3
  28. 28.What is the units digit of 4^44? (a) 4 (b) 6 (c) 7 (d) 5
  29. 29.What is the units digit of 7^50? (a) 1 (b) 7 (c) 9 (d) 3
  30. 30.What is the smallest number you must add to 218 to make it divisible by 8? (a) 7 (b) 2 (c) 5 (d) 6
  31. 31.What is the units digit of 3^28? (a) 3 (b) 1 (c) 7 (d) 9
  32. 32.Which of these numbers is divisible by 6? (a) 1650 (b) 1645 (c) 1658 (d) 1651
  33. 33.What is the smallest number you must add to 207 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  34. 34.What is the units digit of 3^38? (a) 1 (b) 9 (c) 7 (d) 3
  35. 35.What is the units digit of 8^28? (a) 8 (b) 4 (c) 6 (d) 2
  36. 36.Which of these numbers is divisible by 3? (a) 1221 (b) 1226 (c) 1223 (d) 1219
  37. 37.What is the smallest number you must add to 457 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
  38. 38.Which of these numbers is divisible by 3? (a) 1075 (b) 1072 (c) 1071 (d) 1069
  39. 39.What is the smallest number you must add to 139 to make it divisible by 10? (a) 1 (b) 9 (c) 3 (d) 2
  40. 40.What is the smallest number you must add to 280 to make it divisible by 9? (a) 1 (b) 8 (c) 7 (d) 6

Answer key

  1. 1452
  2. 3
  3. 8
  4. 1
  5. 1794
  6. 4
  7. 1536
  8. 5
  9. 4
  10. 2
  11. 1
  12. 6
  13. 3
  14. 4
  15. 3
  16. 1
  17. 7
  18. 5
  19. 8
  20. 3
  21. 540
  22. 3
  23. 7
  24. 4
  25. 3
  26. 3
  27. 2
  28. 6
  29. 9
  30. 6
  31. 1
  32. 1650
  33. 3
  34. 9
  35. 6
  36. 1221
  37. 5
  38. 1071
  39. 1
  40. 8

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