← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 378 · 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 6? (a) 3061 (b) 3076 (c) 3068 (d) 3066
  2. 2.What is the units digit of 8^15? (a) 6 (b) 8 (c) 4 (d) 2
  3. 3.Which of these numbers is divisible by 8? (a) 4747 (b) 4767 (c) 4755 (d) 4752
  4. 4.What is the smallest number you must add to 317 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
  5. 5.Which of these numbers is divisible by 11? (a) 4808 (b) 4804 (c) 4796 (d) 4794
  6. 6.What is the smallest number you must add to 431 to make it divisible by 12? (a) 11 (b) 1 (c) 2 (d) 3
  7. 7.What is the remainder when 271 is divided by 9? (a) 0 (b) 1 (c) 2 (d) 8
  8. 8.Which of these numbers is divisible by 6? (a) 5170 (b) 5156 (c) 5165 (d) 5160
  9. 9.What is the remainder when 374 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  10. 10.What is the remainder when 258 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  11. 11.What is the remainder when 97 is divided by 8? (a) 2 (b) 0 (c) 1 (d) 7
  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 7^27? (a) 1 (b) 3 (c) 7 (d) 9
  14. 14.Which of these numbers is divisible by 8? (a) 5978 (b) 5989 (c) 5984 (d) 5993
  15. 15.What is the units digit of 3^13? (a) 9 (b) 1 (c) 7 (d) 3
  16. 16.What is the smallest number you must add to 299 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 2
  17. 17.What is the smallest number you must add to 366 to make it divisible by 9? (a) 6 (b) 3 (c) 2 (d) 4
  18. 18.What is the units digit of 2^21? (a) 8 (b) 6 (c) 4 (d) 2
  19. 19.Which of these numbers is divisible by 3? (a) 818 (b) 816 (c) 815 (d) 820
  20. 20.What is the smallest number you must add to 164 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
  21. 21.What is the smallest number you must add to 234 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
  22. 22.What is the remainder when 95 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 3
  23. 23.What is the smallest number you must add to 203 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
  24. 24.What is the units digit of 4^21? (a) 5 (b) 3 (c) 4 (d) 6
  25. 25.What is the smallest number you must add to 157 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 6
  26. 26.What is the smallest number you must add to 377 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  27. 27.What is the remainder when 340 is divided by 12? (a) 8 (b) 4 (c) 3 (d) 5
  28. 28.Which of these numbers is divisible by 6? (a) 2083 (b) 2098 (c) 2088 (d) 2091
  29. 29.What is the units digit of 3^59? (a) 3 (b) 9 (c) 7 (d) 1
  30. 30.Which of these numbers is divisible by 3? (a) 1779 (b) 1784 (c) 1781 (d) 1778
  31. 31.What is the smallest number you must add to 491 to make it divisible by 12? (a) 11 (b) 1 (c) 3 (d) 2
  32. 32.What is the remainder when 541 is divided by 11? (a) 2 (b) 3 (c) 1 (d) 9
  33. 33.What is the smallest number you must add to 486 to make it divisible by 11? (a) 8 (b) 9 (c) 2 (d) 10
  34. 34.Which of these numbers is divisible by 8? (a) 3581 (b) 3586 (c) 3598 (d) 3584
  35. 35.Which of these numbers is divisible by 11? (a) 9735 (b) 9742 (c) 9727 (d) 9750
  36. 36.Which of these numbers is divisible by 3? (a) 352 (b) 350 (c) 351 (d) 355
  37. 37.What is the smallest number you must add to 482 to make it divisible by 12? (a) 10 (b) 9 (c) 2 (d) 11
  38. 38.What is the remainder when 714 is divided by 12? (a) 7 (b) 5 (c) 8 (d) 6
  39. 39.Which of these numbers is divisible by 3? (a) 1985 (b) 1982 (c) 1983 (d) 1987
  40. 40.What is the smallest number you must add to 515 to make it divisible by 12? (a) 2 (b) 11 (c) 3 (d) 1

Answer key

  1. 3066
  2. 2
  3. 4752
  4. 5
  5. 4796
  6. 1
  7. 1
  8. 5160
  9. 4
  10. 2
  11. 1
  12. 2
  13. 3
  14. 5984
  15. 3
  16. 1
  17. 3
  18. 2
  19. 816
  20. 4
  21. 6
  22. 4
  23. 5
  24. 4
  25. 4
  26. 3
  27. 4
  28. 2088
  29. 7
  30. 1779
  31. 1
  32. 2
  33. 9
  34. 3584
  35. 9735
  36. 351
  37. 10
  38. 6
  39. 1983
  40. 1

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