← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 728 · 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 smallest number you must add to 127 to make it divisible by 7? (a) 5 (b) 1 (c) 6 (d) 4
  2. 2.What is the smallest number you must add to 396 to make it divisible by 7? (a) 2 (b) 5 (c) 4 (d) 3
  3. 3.What is the smallest number you must add to 221 to make it divisible by 10? (a) 1 (b) 7 (c) 9 (d) 8
  4. 4.What is the units digit of 8^22? (a) 2 (b) 8 (c) 4 (d) 6
  5. 5.What is the smallest number you must add to 548 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 8
  6. 6.What is the units digit of 8^60? (a) 2 (b) 4 (c) 6 (d) 8
  7. 7.Which of these numbers is divisible by 3? (a) 1455 (b) 1459 (c) 1457 (d) 1454
  8. 8.What is the units digit of 2^20? (a) 4 (b) 6 (c) 2 (d) 8
  9. 9.What is the units digit of 2^44? (a) 2 (b) 4 (c) 8 (d) 6
  10. 10.Which of these numbers is divisible by 3? (a) 1760 (b) 1756 (c) 1753 (d) 1755
  11. 11.What is the remainder when 461 is divided by 11? (a) 8 (b) 1 (c) 9 (d) 10
  12. 12.What is the units digit of 7^13? (a) 3 (b) 9 (c) 1 (d) 7
  13. 13.What is the remainder when 538 is divided by 8? (a) 3 (b) 2 (c) 6 (d) 1
  14. 14.What is the units digit of 9^16? (a) 1 (b) 9 (c) 2 (d) 0
  15. 15.What is the units digit of 3^36? (a) 1 (b) 3 (c) 9 (d) 7
  16. 16.What is the units digit of 4^14? (a) 7 (b) 4 (c) 5 (d) 6
  17. 17.What is the smallest number you must add to 459 to make it divisible by 10? (a) 9 (b) 1 (c) 3 (d) 2
  18. 18.What is the smallest number you must add to 623 to make it divisible by 9? (a) 8 (b) 2 (c) 6 (d) 7
  19. 19.What is the smallest number you must add to 435 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
  20. 20.What is the units digit of 4^35? (a) 6 (b) 4 (c) 3 (d) 5
  21. 21.What is the smallest number you must add to 524 to make it divisible by 11? (a) 3 (b) 5 (c) 4 (d) 7
  22. 22.Which of these numbers is divisible by 3? (a) 1826 (b) 1819 (c) 1821 (d) 1823
  23. 23.Which of these numbers is divisible by 11? (a) 4984 (b) 4973 (c) 4972 (d) 4964
  24. 24.Which of these numbers is divisible by 9? (a) 2335 (b) 2341 (c) 2329 (d) 2331
  25. 25.What is the remainder when 66 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
  26. 26.What is the units digit of 4^43? (a) 3 (b) 5 (c) 6 (d) 4
  27. 27.What is the smallest number you must add to 259 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  28. 28.Which of these numbers is divisible by 9? (a) 680 (b) 666 (c) 667 (d) 662
  29. 29.What is the remainder when 208 is divided by 3? (a) 0 (b) 1 (c) 2
  30. 30.What is the smallest number you must add to 778 to make it divisible by 12? (a) 2 (b) 1 (c) 10 (d) 3
  31. 31.Which of these numbers is divisible by 3? (a) 1015 (b) 1021 (c) 1018 (d) 1017
  32. 32.What is the remainder when 828 is divided by 11? (a) 4 (b) 8 (c) 3 (d) 2
  33. 33.What is the smallest number you must add to 702 to make it divisible by 12? (a) 6 (b) 8 (c) 5 (d) 7
  34. 34.Which of these numbers is divisible by 9? (a) 3844 (b) 3839 (c) 3843 (d) 3857
  35. 35.What is the remainder when 148 is divided by 11? (a) 4 (b) 7 (c) 6 (d) 5
  36. 36.What is the remainder when 216 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
  37. 37.What is the smallest number you must add to 302 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  38. 38.What is the remainder when 286 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
  39. 39.What is the smallest number you must add to 406 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 1
  40. 40.What is the smallest number you must add to 56 to make it divisible by 3? (a) 2 (b) 1

Answer key

  1. 6
  2. 3
  3. 9
  4. 4
  5. 1
  6. 6
  7. 1455
  8. 6
  9. 6
  10. 1755
  11. 10
  12. 7
  13. 2
  14. 1
  15. 1
  16. 6
  17. 1
  18. 7
  19. 5
  20. 4
  21. 4
  22. 1821
  23. 4972
  24. 2331
  25. 1
  26. 4
  27. 1
  28. 666
  29. 1
  30. 2
  31. 1017
  32. 3
  33. 6
  34. 3843
  35. 5
  36. 1
  37. 2
  38. 4
  39. 2
  40. 1

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