← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 877 · 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) 2275 (b) 2279 (c) 2274 (d) 2273
  2. 2.What is the remainder when 419 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
  3. 3.What is the remainder when 722 is divided by 11? (a) 7 (b) 6 (c) 8 (d) 4
  4. 4.What is the remainder when 651 is divided by 9? (a) 6 (b) 2 (c) 3 (d) 4
  5. 5.What is the remainder when 519 is divided by 8? (a) 7 (b) 6 (c) 1 (d) 5
  6. 6.What is the smallest number you must add to 911 to make it divisible by 12? (a) 3 (b) 1 (c) 2 (d) 11
  7. 7.What is the smallest number you must add to 209 to make it divisible by 9? (a) 7 (b) 2 (c) 6 (d) 8
  8. 8.What is the smallest number you must add to 412 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 6
  9. 9.What is the smallest number you must add to 347 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 2
  10. 10.What is the units digit of 3^18? (a) 9 (b) 3 (c) 7 (d) 1
  11. 11.What is the smallest number you must add to 161 to make it divisible by 3? (a) 1 (b) 2
  12. 12.Which of these numbers is divisible by 6? (a) 3756 (b) 3761 (c) 3766 (d) 3754
  13. 13.What is the units digit of 9^22? (a) 2 (b) 9 (c) 1 (d) 0
  14. 14.What is the remainder when 588 is divided by 10? (a) 2 (b) 9 (c) 8 (d) 7
  15. 15.Which of these numbers is divisible by 11? (a) 1110 (b) 1100 (c) 1121 (d) 1091
  16. 16.Which of these numbers is divisible by 6? (a) 5226 (b) 5234 (c) 5229 (d) 5223
  17. 17.What is the remainder when 529 is divided by 8? (a) 0 (b) 1 (c) 2 (d) 7
  18. 18.Which of these numbers is divisible by 4? (a) 2118 (b) 2114 (c) 2111 (d) 2112
  19. 19.What is the smallest number you must add to 142 to make it divisible by 9? (a) 2 (b) 3 (c) 1 (d) 7
  20. 20.What is the smallest number you must add to 125 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
  21. 21.What is the remainder when 444 is divided by 9? (a) 6 (b) 3 (c) 4 (d) 2
  22. 22.What is the remainder when 235 is divided by 10? (a) 6 (b) 5 (c) 7 (d) 4
  23. 23.What is the units digit of 3^59? (a) 3 (b) 1 (c) 9 (d) 7
  24. 24.What is the smallest number you must add to 32 to make it divisible by 3? (a) 2 (b) 1
  25. 25.Which of these numbers is divisible by 4? (a) 1712 (b) 1718 (c) 1711 (d) 1715
  26. 26.Which of these numbers is divisible by 8? (a) 5506 (b) 5516 (c) 5512 (d) 5525
  27. 27.What is the remainder when 114 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  28. 28.What is the smallest number you must add to 344 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
  29. 29.What is the remainder when 289 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
  30. 30.Which of these numbers is divisible by 6? (a) 1771 (b) 1764 (c) 1765 (d) 1760
  31. 31.Which of these numbers is divisible by 8? (a) 631 (b) 618 (c) 633 (d) 624
  32. 32.Which of these numbers is divisible by 6? (a) 991 (b) 982 (c) 984 (d) 989
  33. 33.What is the smallest number you must add to 377 to make it divisible by 7? (a) 6 (b) 2 (c) 3 (d) 1
  34. 34.What is the remainder when 137 is divided by 3? (a) 1 (b) 0 (c) 2
  35. 35.What is the smallest number you must add to 111 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  36. 36.What is the smallest number you must add to 456 to make it divisible by 9? (a) 4 (b) 3 (c) 2 (d) 6
  37. 37.What is the smallest number you must add to 235 to make it divisible by 3? (a) 1 (b) 2
  38. 38.What is the smallest number you must add to 104 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  39. 39.What is the remainder when 242 is divided by 10? (a) 8 (b) 2 (c) 1 (d) 3
  40. 40.What is the remainder when 47 is divided by 3? (a) 1 (b) 2 (c) 0

Answer key

  1. 2274
  2. 5
  3. 7
  4. 3
  5. 7
  6. 1
  7. 7
  8. 1
  9. 1
  10. 9
  11. 1
  12. 3756
  13. 1
  14. 8
  15. 1100
  16. 5226
  17. 1
  18. 2112
  19. 2
  20. 1
  21. 3
  22. 5
  23. 7
  24. 1
  25. 1712
  26. 5512
  27. 4
  28. 6
  29. 4
  30. 1764
  31. 624
  32. 984
  33. 1
  34. 2
  35. 4
  36. 3
  37. 2
  38. 1
  39. 2
  40. 2

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad