← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 401 · 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 9? (a) 1234 (b) 1233 (c) 1230 (d) 1245
  2. 2.What is the remainder when 123 is divided by 12? (a) 2 (b) 4 (c) 9 (d) 3
  3. 3.What is the remainder when 50 is divided by 3? (a) 2 (b) 0 (c) 1
  4. 4.Which of these numbers is divisible by 9? (a) 7906 (b) 7918 (c) 7896 (d) 7902
  5. 5.What is the units digit of 2^40? (a) 2 (b) 6 (c) 8 (d) 4
  6. 6.What is the remainder when 67 is divided by 3? (a) 0 (b) 1 (c) 2
  7. 7.What is the remainder when 557 is divided by 11? (a) 8 (b) 4 (c) 6 (d) 7
  8. 8.What is the smallest number you must add to 304 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
  9. 9.Which of these numbers is divisible by 11? (a) 1518 (b) 1534 (c) 1527 (d) 1517
  10. 10.What is the remainder when 326 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 6
  11. 11.What is the units digit of 9^56? (a) 0 (b) 1 (c) 2 (d) 9
  12. 12.What is the remainder when 57 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
  13. 13.What is the smallest number you must add to 381 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  14. 14.Which of these numbers is divisible by 4? (a) 2286 (b) 2288 (c) 2291 (d) 2294
  15. 15.What is the units digit of 8^25? (a) 4 (b) 8 (c) 6 (d) 2
  16. 16.What is the units digit of 4^33? (a) 5 (b) 3 (c) 6 (d) 4
  17. 17.What is the remainder when 299 is divided by 10? (a) 8 (b) 1 (c) 7 (d) 9
  18. 18.What is the smallest number you must add to 268 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  19. 19.Which of these numbers is divisible by 4? (a) 2678 (b) 2674 (c) 2672 (d) 2670
  20. 20.What is the remainder when 321 is divided by 7? (a) 4 (b) 1 (c) 5 (d) 6
  21. 21.What is the units digit of 9^13? (a) 8 (b) 7 (c) 9 (d) 1
  22. 22.What is the units digit of 9^14? (a) 1 (b) 0 (c) 2 (d) 9
  23. 23.What is the smallest number you must add to 249 to make it divisible by 8? (a) 7 (b) 6 (c) 1 (d) 5
  24. 24.What is the remainder when 94 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  25. 25.What is the smallest number you must add to 259 to make it divisible by 12? (a) 6 (b) 4 (c) 5 (d) 7
  26. 26.What is the smallest number you must add to 71 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  27. 27.What is the units digit of 7^25? (a) 7 (b) 3 (c) 9 (d) 1
  28. 28.What is the smallest number you must add to 546 to make it divisible by 10? (a) 5 (b) 6 (c) 3 (d) 4
  29. 29.What is the units digit of 3^58? (a) 7 (b) 9 (c) 3 (d) 1
  30. 30.Which of these numbers is divisible by 6? (a) 1776 (b) 1779 (c) 1787 (d) 1773
  31. 31.What is the smallest number you must add to 236 to make it divisible by 3? (a) 2 (b) 1
  32. 32.What is the smallest number you must add to 199 to make it divisible by 12? (a) 6 (b) 7 (c) 4 (d) 5
  33. 33.What is the smallest number you must add to 452 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6
  34. 34.Which of these numbers is divisible by 8? (a) 4556 (b) 4560 (c) 4570 (d) 4566
  35. 35.What is the smallest number you must add to 106 to make it divisible by 3? (a) 2 (b) 1
  36. 36.What is the units digit of 2^51? (a) 6 (b) 2 (c) 4 (d) 8
  37. 37.What is the smallest number you must add to 374 to make it divisible by 7? (a) 6 (b) 4 (c) 3 (d) 5
  38. 38.What is the remainder when 211 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
  39. 39.What is the smallest number you must add to 853 to make it divisible by 11? (a) 4 (b) 5 (c) 6 (d) 7
  40. 40.What is the remainder when 224 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7

Answer key

  1. 1233
  2. 3
  3. 2
  4. 7902
  5. 6
  6. 1
  7. 7
  8. 6
  9. 1518
  10. 4
  11. 1
  12. 1
  13. 4
  14. 2288
  15. 8
  16. 4
  17. 9
  18. 2
  19. 2672
  20. 6
  21. 9
  22. 1
  23. 7
  24. 4
  25. 5
  26. 4
  27. 7
  28. 4
  29. 9
  30. 1776
  31. 1
  32. 5
  33. 7
  34. 4560
  35. 2
  36. 8
  37. 4
  38. 1
  39. 5
  40. 8

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