← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 319 · 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 3^32? (a) 9 (b) 7 (c) 1 (d) 3
  2. 2.Which of these numbers is divisible by 9? (a) 2698 (b) 2701 (c) 2691 (d) 2690
  3. 3.What is the smallest number you must add to 332 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
  4. 4.What is the units digit of 3^22? (a) 1 (b) 3 (c) 9 (d) 7
  5. 5.What is the smallest number you must add to 225 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  6. 6.What is the smallest number you must add to 86 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  7. 7.What is the smallest number you must add to 369 to make it divisible by 12? (a) 9 (b) 3 (c) 4 (d) 2
  8. 8.Which of these numbers is divisible by 11? (a) 2750 (b) 2770 (c) 2754 (d) 2747
  9. 9.What is the units digit of 8^41? (a) 8 (b) 2 (c) 4 (d) 6
  10. 10.What is the units digit of 3^60? (a) 3 (b) 9 (c) 7 (d) 1
  11. 11.What is the units digit of 4^21? (a) 5 (b) 6 (c) 3 (d) 4
  12. 12.What is the smallest number you must add to 53 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  13. 13.Which of these numbers is divisible by 8? (a) 4459 (b) 4452 (c) 4465 (d) 4456
  14. 14.What is the smallest number you must add to 299 to make it divisible by 11? (a) 9 (b) 2 (c) 10 (d) 8
  15. 15.What is the units digit of 2^15? (a) 2 (b) 8 (c) 4 (d) 6
  16. 16.What is the units digit of 8^43? (a) 8 (b) 6 (c) 2 (d) 4
  17. 17.Which of these numbers is divisible by 6? (a) 1090 (b) 1086 (c) 1082 (d) 1096
  18. 18.What is the units digit of 2^55? (a) 6 (b) 4 (c) 2 (d) 8
  19. 19.What is the smallest number you must add to 751 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 4
  20. 20.What is the remainder when 889 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
  21. 21.Which of these numbers is divisible by 4? (a) 956 (b) 961 (c) 957 (d) 953
  22. 22.Which of these numbers is divisible by 4? (a) 1599 (b) 1603 (c) 1606 (d) 1600
  23. 23.What is the smallest number you must add to 510 to make it divisible by 7? (a) 6 (b) 2 (c) 1 (d) 3
  24. 24.What is the remainder when 115 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
  25. 25.What is the remainder when 421 is divided by 8? (a) 5 (b) 6 (c) 4 (d) 3
  26. 26.What is the units digit of 2^27? (a) 6 (b) 8 (c) 2 (d) 4
  27. 27.What is the units digit of 4^30? (a) 5 (b) 7 (c) 4 (d) 6
  28. 28.Which of these numbers is divisible by 6? (a) 1884 (b) 1882 (c) 1888 (d) 1891
  29. 29.What is the smallest number you must add to 172 to make it divisible by 7? (a) 3 (b) 2 (c) 4 (d) 5
  30. 30.Which of these numbers is divisible by 11? (a) 904 (b) 899 (c) 902 (d) 920
  31. 31.What is the remainder when 374 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  32. 32.What is the units digit of 3^47? (a) 1 (b) 9 (c) 7 (d) 3
  33. 33.What is the remainder when 309 is divided by 7? (a) 0 (b) 6 (c) 2 (d) 1
  34. 34.What is the units digit of 7^55? (a) 9 (b) 3 (c) 7 (d) 1
  35. 35.What is the smallest number you must add to 127 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 1
  36. 36.What is the units digit of 9^38? (a) 1 (b) 9 (c) 0 (d) 2
  37. 37.What is the units digit of 3^51? (a) 7 (b) 1 (c) 3 (d) 9
  38. 38.What is the smallest number you must add to 97 to make it divisible by 3? (a) 2 (b) 1
  39. 39.What is the units digit of 8^39? (a) 6 (b) 8 (c) 4 (d) 2
  40. 40.What is the smallest number you must add to 434 to make it divisible by 9? (a) 6 (b) 7 (c) 2 (d) 8

Answer key

  1. 1
  2. 2691
  3. 4
  4. 9
  5. 3
  6. 2
  7. 3
  8. 2750
  9. 8
  10. 1
  11. 4
  12. 3
  13. 4456
  14. 9
  15. 8
  16. 2
  17. 1086
  18. 8
  19. 5
  20. 9
  21. 956
  22. 1600
  23. 1
  24. 3
  25. 5
  26. 8
  27. 6
  28. 1884
  29. 3
  30. 902
  31. 4
  32. 7
  33. 1
  34. 3
  35. 8
  36. 1
  37. 7
  38. 2
  39. 2
  40. 7

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