← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 202 · 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 726 to make it divisible by 12? (a) 7 (b) 5 (c) 8 (d) 6
  2. 2.What is the remainder when 386 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
  3. 3.What is the units digit of 2^15? (a) 4 (b) 8 (c) 6 (d) 2
  4. 4.Which of these numbers is divisible by 6? (a) 2876 (b) 2873 (c) 2883 (d) 2874
  5. 5.What is the remainder when 97 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
  6. 6.What is the smallest number you must add to 238 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  7. 7.What is the smallest number you must add to 621 to make it divisible by 8? (a) 2 (b) 3 (c) 4 (d) 5
  8. 8.What is the smallest number you must add to 138 to make it divisible by 7? (a) 2 (b) 1 (c) 3 (d) 5
  9. 9.Which of these numbers is divisible by 4? (a) 456 (b) 459 (c) 463 (d) 453
  10. 10.Which of these numbers is divisible by 3? (a) 1762 (b) 1756 (c) 1758 (d) 1760
  11. 11.What is the smallest number you must add to 158 to make it divisible by 3? (a) 1 (b) 2
  12. 12.Which of these numbers is divisible by 3? (a) 1535 (b) 1529 (c) 1530 (d) 1532
  13. 13.Which of these numbers is divisible by 4? (a) 914 (b) 907 (c) 911 (d) 908
  14. 14.What is the units digit of 3^49? (a) 7 (b) 3 (c) 9 (d) 1
  15. 15.Which of these numbers is divisible by 9? (a) 6331 (b) 6344 (c) 6348 (d) 6336
  16. 16.What is the smallest number you must add to 514 to make it divisible by 7? (a) 5 (b) 4 (c) 3 (d) 6
  17. 17.What is the smallest number you must add to 31 to make it divisible by 3? (a) 1 (b) 2
  18. 18.What is the remainder when 280 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
  19. 19.What is the smallest number you must add to 302 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  20. 20.Which of these numbers is divisible by 3? (a) 2448 (b) 2447 (c) 2453 (d) 2449
  21. 21.What is the remainder when 647 is divided by 9? (a) 1 (b) 7 (c) 8 (d) 6
  22. 22.What is the smallest number you must add to 587 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6
  23. 23.What is the smallest number you must add to 274 to make it divisible by 12? (a) 10 (b) 2 (c) 1 (d) 3
  24. 24.Which of these numbers is divisible by 3? (a) 2297 (b) 2293 (c) 2300 (d) 2295
  25. 25.What is the smallest number you must add to 87 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  26. 26.What is the remainder when 147 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
  27. 27.Which of these numbers is divisible by 3? (a) 2440 (b) 2443 (c) 2437 (d) 2439
  28. 28.What is the smallest number you must add to 103 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  29. 29.What is the units digit of 3^33? (a) 7 (b) 1 (c) 3 (d) 9
  30. 30.What is the smallest number you must add to 317 to make it divisible by 9? (a) 6 (b) 8 (c) 2 (d) 7
  31. 31.Which of these numbers is divisible by 9? (a) 6464 (b) 6471 (c) 6485 (d) 6473
  32. 32.What is the smallest number you must add to 385 to make it divisible by 12? (a) 1 (b) 9 (c) 11 (d) 10
  33. 33.Which of these numbers is divisible by 9? (a) 2691 (b) 2684 (c) 2696 (d) 2708
  34. 34.What is the remainder when 217 is divided by 3? (a) 2 (b) 0 (c) 1
  35. 35.What is the units digit of 8^30? (a) 6 (b) 2 (c) 4 (d) 8
  36. 36.Which of these numbers is divisible by 6? (a) 2701 (b) 2707 (c) 2706 (d) 2717
  37. 37.What is the smallest number you must add to 550 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 10
  38. 38.What is the smallest number you must add to 257 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  39. 39.Which of these numbers is divisible by 3? (a) 1497 (b) 1499 (c) 1496 (d) 1501
  40. 40.What is the remainder when 358 is divided by 7? (a) 6 (b) 1 (c) 0 (d) 2

Answer key

  1. 6
  2. 6
  3. 8
  4. 2874
  5. 1
  6. 2
  7. 3
  8. 2
  9. 456
  10. 1758
  11. 1
  12. 1530
  13. 908
  14. 3
  15. 6336
  16. 4
  17. 2
  18. 4
  19. 2
  20. 2448
  21. 8
  22. 7
  23. 2
  24. 2295
  25. 1
  26. 3
  27. 2439
  28. 2
  29. 3
  30. 7
  31. 6471
  32. 11
  33. 2691
  34. 1
  35. 4
  36. 2706
  37. 2
  38. 3
  39. 1497
  40. 1

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