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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 303 · 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 167 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  2. 2.What is the smallest number you must add to 170 to make it divisible by 7? (a) 6 (b) 4 (c) 2 (d) 5
  3. 3.What is the smallest number you must add to 424 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
  4. 4.What is the remainder when 561 is divided by 8? (a) 2 (b) 0 (c) 1 (d) 7
  5. 5.Which of these numbers is divisible by 6? (a) 395 (b) 382 (c) 384 (d) 386
  6. 6.What is the units digit of 2^50? (a) 8 (b) 2 (c) 4 (d) 6
  7. 7.Which of these numbers is divisible by 8? (a) 1582 (b) 1569 (c) 1568 (d) 1563
  8. 8.What is the remainder when 418 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
  9. 9.What is the units digit of 2^47? (a) 4 (b) 8 (c) 2 (d) 6
  10. 10.What is the units digit of 4^20? (a) 4 (b) 6 (c) 7 (d) 5
  11. 11.What is the remainder when 893 is divided by 12? (a) 4 (b) 5 (c) 7 (d) 6
  12. 12.What is the units digit of 9^43? (a) 1 (b) 9 (c) 7 (d) 8
  13. 13.What is the remainder when 45 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  14. 14.Which of these numbers is divisible by 8? (a) 4449 (b) 4448 (c) 4447 (d) 4458
  15. 15.What is the remainder when 471 is divided by 10? (a) 1 (b) 0 (c) 2 (d) 9
  16. 16.Which of these numbers is divisible by 11? (a) 1854 (b) 1828 (c) 1837 (d) 1838
  17. 17.Which of these numbers is divisible by 6? (a) 2039 (b) 2040 (c) 2049 (d) 2043
  18. 18.What is the units digit of 2^33? (a) 4 (b) 2 (c) 6 (d) 8
  19. 19.What is the remainder when 314 is divided by 6? (a) 3 (b) 2 (c) 1 (d) 4
  20. 20.What is the smallest number you must add to 848 to make it divisible by 12? (a) 3 (b) 5 (c) 4 (d) 8
  21. 21.What is the remainder when 221 is divided by 12? (a) 7 (b) 4 (c) 5 (d) 6
  22. 22.What is the remainder when 492 is divided by 10? (a) 8 (b) 3 (c) 2 (d) 1
  23. 23.What is the smallest number you must add to 917 to make it divisible by 12? (a) 8 (b) 6 (c) 7 (d) 5
  24. 24.What is the smallest number you must add to 218 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  25. 25.What is the units digit of 8^36? (a) 8 (b) 4 (c) 6 (d) 2
  26. 26.What is the remainder when 267 is divided by 12? (a) 4 (b) 3 (c) 9 (d) 2
  27. 27.What is the units digit of 7^51? (a) 7 (b) 3 (c) 9 (d) 1
  28. 28.What is the smallest number you must add to 544 to make it divisible by 7? (a) 2 (b) 1 (c) 5 (d) 3
  29. 29.Which of these numbers is divisible by 4? (a) 2635 (b) 2637 (c) 2641 (d) 2636
  30. 30.What is the smallest number you must add to 389 to make it divisible by 8? (a) 5 (b) 3 (c) 2 (d) 4
  31. 31.What is the units digit of 2^11? (a) 4 (b) 8 (c) 6 (d) 2
  32. 32.What is the smallest number you must add to 339 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 3
  33. 33.What is the units digit of 8^29? (a) 8 (b) 2 (c) 6 (d) 4
  34. 34.Which of these numbers is divisible by 6? (a) 3797 (b) 3789 (c) 3781 (d) 3786
  35. 35.What is the remainder when 250 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
  36. 36.What is the units digit of 2^32? (a) 8 (b) 4 (c) 6 (d) 2
  37. 37.What is the remainder when 232 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
  38. 38.What is the units digit of 3^51? (a) 3 (b) 1 (c) 9 (d) 7
  39. 39.What is the units digit of 3^58? (a) 9 (b) 1 (c) 3 (d) 7
  40. 40.What is the remainder when 510 is divided by 11? (a) 5 (b) 4 (c) 3 (d) 7

Answer key

  1. 1
  2. 5
  3. 8
  4. 1
  5. 384
  6. 4
  7. 1568
  8. 2
  9. 8
  10. 6
  11. 5
  12. 9
  13. 1
  14. 4448
  15. 1
  16. 1837
  17. 2040
  18. 2
  19. 2
  20. 4
  21. 5
  22. 2
  23. 7
  24. 2
  25. 6
  26. 3
  27. 3
  28. 2
  29. 2636
  30. 3
  31. 8
  32. 4
  33. 8
  34. 3786
  35. 4
  36. 6
  37. 1
  38. 7
  39. 9
  40. 4

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