← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 749 · 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 remainder when 103 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  2. 2.What is the smallest number you must add to 433 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 1
  3. 3.What is the remainder when 211 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
  4. 4.What is the units digit of 2^19? (a) 2 (b) 4 (c) 6 (d) 8
  5. 5.What is the smallest number you must add to 718 to make it divisible by 11? (a) 8 (b) 3 (c) 7 (d) 9
  6. 6.What is the remainder when 188 is divided by 3? (a) 2 (b) 0 (c) 1
  7. 7.What is the smallest number you must add to 633 to make it divisible by 8? (a) 6 (b) 1 (c) 5 (d) 7
  8. 8.What is the remainder when 769 is divided by 12? (a) 1 (b) 2 (c) 11 (d) 0
  9. 9.What is the remainder when 748 is divided by 12? (a) 8 (b) 4 (c) 3 (d) 5
  10. 10.What is the smallest number you must add to 408 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 3
  11. 11.What is the remainder when 242 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
  12. 12.What is the remainder when 671 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 4
  13. 13.What is the units digit of 2^24? (a) 6 (b) 4 (c) 2 (d) 8
  14. 14.Which of these numbers is divisible by 4? (a) 2940 (b) 2938 (c) 2941 (d) 2945
  15. 15.What is the remainder when 147 is divided by 9? (a) 2 (b) 6 (c) 4 (d) 3
  16. 16.What is the units digit of 4^54? (a) 4 (b) 7 (c) 5 (d) 6
  17. 17.Which of these numbers is divisible by 4? (a) 3541 (b) 3537 (c) 3534 (d) 3536
  18. 18.Which of these numbers is divisible by 3? (a) 1565 (b) 1560 (c) 1561 (d) 1558
  19. 19.Which of these numbers is divisible by 4? (a) 2971 (b) 2962 (c) 2964 (d) 2966
  20. 20.What is the smallest number you must add to 386 to make it divisible by 11? (a) 1 (b) 9 (c) 10 (d) 8
  21. 21.What is the remainder when 340 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 3
  22. 22.What is the remainder when 450 is divided by 7? (a) 3 (b) 1 (c) 5 (d) 2
  23. 23.Which of these numbers is divisible by 3? (a) 1229 (b) 1231 (c) 1227 (d) 1225
  24. 24.What is the remainder when 130 is divided by 3? (a) 2 (b) 1 (c) 0
  25. 25.What is the remainder when 186 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
  26. 26.What is the units digit of 7^51? (a) 7 (b) 3 (c) 9 (d) 1
  27. 27.Which of these numbers is divisible by 3? (a) 1643 (b) 1646 (c) 1649 (d) 1644
  28. 28.Which of these numbers is divisible by 8? (a) 4753 (b) 4770 (c) 4764 (d) 4760
  29. 29.What is the smallest number you must add to 178 to make it divisible by 11? (a) 2 (b) 9 (c) 8 (d) 10
  30. 30.What is the units digit of 3^22? (a) 3 (b) 9 (c) 1 (d) 7
  31. 31.What is the smallest number you must add to 488 to make it divisible by 7? (a) 1 (b) 2 (c) 5 (d) 3
  32. 32.Which of these numbers is divisible by 11? (a) 6736 (b) 6720 (c) 6730 (d) 6721
  33. 33.What is the smallest number you must add to 383 to make it divisible by 12? (a) 3 (b) 1 (c) 2 (d) 11
  34. 34.What is the units digit of 4^44? (a) 7 (b) 5 (c) 4 (d) 6
  35. 35.What is the smallest number you must add to 108 to make it divisible by 10? (a) 1 (b) 8 (c) 2 (d) 3
  36. 36.Which of these numbers is divisible by 3? (a) 1832 (b) 1825 (c) 1828 (d) 1827
  37. 37.What is the smallest number you must add to 142 to make it divisible by 11? (a) 10 (b) 2 (c) 1 (d) 3
  38. 38.What is the remainder when 147 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
  39. 39.Which of these numbers is divisible by 3? (a) 2183 (b) 2177 (c) 2178 (d) 2179
  40. 40.What is the smallest number you must add to 315 to make it divisible by 12? (a) 3 (b) 9 (c) 10 (d) 8

Answer key

  1. 3
  2. 11
  3. 1
  4. 8
  5. 8
  6. 2
  7. 7
  8. 1
  9. 4
  10. 6
  11. 2
  12. 5
  13. 6
  14. 2940
  15. 3
  16. 6
  17. 3536
  18. 1560
  19. 2964
  20. 10
  21. 4
  22. 2
  23. 1227
  24. 1
  25. 6
  26. 3
  27. 1644
  28. 4760
  29. 9
  30. 9
  31. 2
  32. 6721
  33. 1
  34. 6
  35. 2
  36. 1827
  37. 1
  38. 3
  39. 2178
  40. 9

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