← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 697 · 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 units digit of 7^53? (a) 3 (b) 1 (c) 9 (d) 7
  2. 2.What is the remainder when 355 is divided by 7? (a) 4 (b) 2 (c) 6 (d) 5
  3. 3.Which of these numbers is divisible by 8? (a) 5101 (b) 5108 (c) 5104 (d) 5118
  4. 4.What is the units digit of 4^54? (a) 4 (b) 6 (c) 5 (d) 7
  5. 5.What is the units digit of 2^41? (a) 6 (b) 4 (c) 8 (d) 2
  6. 6.What is the remainder when 246 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
  7. 7.What is the remainder when 525 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
  8. 8.What is the smallest number you must add to 504 to make it divisible by 11? (a) 9 (b) 1 (c) 3 (d) 2
  9. 9.What is the smallest number you must add to 673 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
  10. 10.What is the units digit of 2^58? (a) 8 (b) 2 (c) 6 (d) 4
  11. 11.Which of these numbers is divisible by 8? (a) 4849 (b) 4859 (c) 4842 (d) 4848
  12. 12.What is the smallest number you must add to 232 to make it divisible by 3? (a) 1 (b) 2
  13. 13.What is the units digit of 4^21? (a) 3 (b) 5 (c) 6 (d) 4
  14. 14.What is the smallest number you must add to 137 to make it divisible by 3? (a) 2 (b) 1
  15. 15.What is the units digit of 3^28? (a) 3 (b) 9 (c) 7 (d) 1
  16. 16.What is the smallest number you must add to 405 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6
  17. 17.What is the units digit of 3^33? (a) 7 (b) 9 (c) 1 (d) 3
  18. 18.What is the remainder when 361 is divided by 6? (a) 2 (b) 1 (c) 0 (d) 5
  19. 19.What is the smallest number you must add to 181 to make it divisible by 3? (a) 1 (b) 2
  20. 20.What is the smallest number you must add to 678 to make it divisible by 11? (a) 7 (b) 5 (c) 4 (d) 3
  21. 21.Which of these numbers is divisible by 11? (a) 5750 (b) 5742 (c) 5739 (d) 5762
  22. 22.What is the smallest number you must add to 135 to make it divisible by 8? (a) 7 (b) 3 (c) 1 (d) 2
  23. 23.What is the smallest number you must add to 148 to make it divisible by 10? (a) 1 (b) 3 (c) 2 (d) 8
  24. 24.What is the smallest number you must add to 203 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  25. 25.Which of these numbers is divisible by 9? (a) 2416 (b) 2410 (c) 2403 (d) 2398
  26. 26.What is the smallest number you must add to 347 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  27. 27.What is the units digit of 8^15? (a) 8 (b) 4 (c) 6 (d) 2
  28. 28.Which of these numbers is divisible by 3? (a) 2019 (b) 2017 (c) 2021 (d) 2023
  29. 29.What is the units digit of 2^36? (a) 6 (b) 2 (c) 8 (d) 4
  30. 30.What is the remainder when 579 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
  31. 31.What is the remainder when 259 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
  32. 32.What is the smallest number you must add to 777 to make it divisible by 10? (a) 4 (b) 2 (c) 7 (d) 3
  33. 33.What is the smallest number you must add to 139 to make it divisible by 3? (a) 1 (b) 2
  34. 34.Which of these numbers is divisible by 4? (a) 1903 (b) 1906 (c) 1899 (d) 1900
  35. 35.Which of these numbers is divisible by 4? (a) 3199 (b) 3193 (c) 3202 (d) 3196
  36. 36.What is the smallest number you must add to 155 to make it divisible by 6? (a) 2 (b) 5 (c) 1 (d) 3
  37. 37.Which of these numbers is divisible by 4? (a) 221 (b) 229 (c) 227 (d) 224
  38. 38.What is the units digit of 7^30? (a) 1 (b) 9 (c) 7 (d) 3
  39. 39.What is the smallest number you must add to 305 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  40. 40.What is the smallest number you must add to 578 to make it divisible by 11? (a) 5 (b) 7 (c) 6 (d) 4

Answer key

  1. 7
  2. 5
  3. 5104
  4. 6
  5. 2
  6. 1
  7. 5
  8. 2
  9. 2
  10. 4
  11. 4848
  12. 2
  13. 4
  14. 1
  15. 1
  16. 1
  17. 3
  18. 1
  19. 2
  20. 4
  21. 5742
  22. 1
  23. 2
  24. 2
  25. 2403
  26. 3
  27. 2
  28. 2019
  29. 6
  30. 3
  31. 3
  32. 3
  33. 2
  34. 1900
  35. 3196
  36. 1
  37. 224
  38. 9
  39. 3
  40. 5

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