← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 820 · 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) 1678 (b) 1676 (c) 1673 (d) 1674
  2. 2.What is the smallest number you must add to 41 to make it divisible by 3? (a) 2 (b) 1
  3. 3.What is the remainder when 232 is divided by 3? (a) 2 (b) 1 (c) 0
  4. 4.What is the remainder when 269 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
  5. 5.What is the remainder when 145 is divided by 8? (a) 0 (b) 1 (c) 7 (d) 2
  6. 6.What is the smallest number you must add to 725 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
  7. 7.What is the remainder when 243 is divided by 10? (a) 3 (b) 2 (c) 7 (d) 4
  8. 8.What is the smallest number you must add to 336 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  9. 9.What is the smallest number you must add to 311 to make it divisible by 11? (a) 9 (b) 7 (c) 3 (d) 8
  10. 10.What is the units digit of 9^54? (a) 0 (b) 2 (c) 1 (d) 9
  11. 11.Which of these numbers is divisible by 3? (a) 2026 (b) 2023 (c) 2030 (d) 2025
  12. 12.What is the smallest number you must add to 262 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  13. 13.Which of these numbers is divisible by 4? (a) 245 (b) 243 (c) 240 (d) 237
  14. 14.What is the units digit of 8^56? (a) 2 (b) 4 (c) 6 (d) 8
  15. 15.What is the remainder when 166 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
  16. 16.What is the smallest number you must add to 227 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5
  17. 17.Which of these numbers is divisible by 9? (a) 4114 (b) 4104 (c) 4099 (d) 4111
  18. 18.What is the smallest number you must add to 160 to make it divisible by 12? (a) 7 (b) 4 (c) 9 (d) 8
  19. 19.What is the remainder when 722 is divided by 9? (a) 3 (b) 7 (c) 2 (d) 1
  20. 20.What is the smallest number you must add to 73 to make it divisible by 7? (a) 4 (b) 6 (c) 3 (d) 5
  21. 21.Which of these numbers is divisible by 11? (a) 7141 (b) 7150 (c) 7163 (d) 7156
  22. 22.What is the smallest number you must add to 512 to make it divisible by 12? (a) 3 (b) 8 (c) 5 (d) 4
  23. 23.What is the smallest number you must add to 50 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  24. 24.What is the units digit of 7^18? (a) 1 (b) 7 (c) 3 (d) 9
  25. 25.Which of these numbers is divisible by 3? (a) 1536 (b) 1540 (c) 1535 (d) 1538
  26. 26.What is the smallest number you must add to 161 to make it divisible by 11? (a) 5 (b) 4 (c) 7 (d) 3
  27. 27.Which of these numbers is divisible by 9? (a) 6666 (b) 6669 (c) 6684 (d) 6673
  28. 28.Which of these numbers is divisible by 9? (a) 783 (b) 782 (c) 791 (d) 798
  29. 29.What is the smallest number you must add to 458 to make it divisible by 8? (a) 7 (b) 5 (c) 2 (d) 6
  30. 30.What is the units digit of 4^16? (a) 4 (b) 7 (c) 6 (d) 5
  31. 31.What is the remainder when 122 is divided by 3? (a) 2 (b) 0 (c) 1
  32. 32.What is the smallest number you must add to 953 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
  33. 33.What is the remainder when 779 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
  34. 34.Which of these numbers is divisible by 6? (a) 3012 (b) 3021 (c) 3013 (d) 3010
  35. 35.What is the remainder when 224 is divided by 11? (a) 3 (b) 4 (c) 7 (d) 5
  36. 36.What is the units digit of 7^31? (a) 7 (b) 3 (c) 9 (d) 1
  37. 37.Which of these numbers is divisible by 8? (a) 1520 (b) 1532 (c) 1515 (d) 1523
  38. 38.What is the remainder when 97 is divided by 3? (a) 0 (b) 2 (c) 1
  39. 39.Which of these numbers is divisible by 11? (a) 1491 (b) 1497 (c) 1478 (d) 1485
  40. 40.What is the remainder when 273 is divided by 10? (a) 3 (b) 7 (c) 2 (d) 4

Answer key

  1. 1674
  2. 1
  3. 1
  4. 5
  5. 1
  6. 4
  7. 3
  8. 4
  9. 8
  10. 1
  11. 2025
  12. 2
  13. 240
  14. 6
  15. 2
  16. 5
  17. 4104
  18. 8
  19. 2
  20. 4
  21. 7150
  22. 4
  23. 2
  24. 9
  25. 1536
  26. 4
  27. 6669
  28. 783
  29. 6
  30. 6
  31. 2
  32. 7
  33. 9
  34. 3012
  35. 4
  36. 3
  37. 1520
  38. 1
  39. 1485
  40. 3

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