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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 18 · 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) 1389 (b) 1392 (c) 1395 (d) 1400
  2. 2.What is the remainder when 124 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  3. 3.What is the smallest number you must add to 326 to make it divisible by 12? (a) 2 (b) 9 (c) 10 (d) 11
  4. 4.Which of these numbers is divisible by 3? (a) 437 (b) 431 (c) 432 (d) 433
  5. 5.What is the smallest number you must add to 65 to make it divisible by 3? (a) 2 (b) 1
  6. 6.Which of these numbers is divisible by 9? (a) 5481 (b) 5493 (c) 5477 (d) 5482
  7. 7.What is the smallest number you must add to 788 to make it divisible by 11? (a) 5 (b) 3 (c) 4 (d) 7
  8. 8.What is the remainder when 221 is divided by 3? (a) 0 (b) 1 (c) 2
  9. 9.What is the smallest number you must add to 205 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 6
  10. 10.What is the units digit of 4^11? (a) 4 (b) 3 (c) 5 (d) 6
  11. 11.Which of these numbers is divisible by 3? (a) 1922 (b) 1915 (c) 1917 (d) 1918
  12. 12.What is the smallest number you must add to 508 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 2
  13. 13.Which of these numbers is divisible by 3? (a) 2422 (b) 2421 (c) 2425 (d) 2420
  14. 14.Which of these numbers is divisible by 8? (a) 3545 (b) 3542 (c) 3533 (d) 3536
  15. 15.Which of these numbers is divisible by 9? (a) 5080 (b) 5076 (c) 5075 (d) 5090
  16. 16.What is the units digit of 2^19? (a) 4 (b) 2 (c) 6 (d) 8
  17. 17.Which of these numbers is divisible by 4? (a) 1763 (b) 1753 (c) 1758 (d) 1756
  18. 18.What is the remainder when 442 is divided by 11? (a) 1 (b) 3 (c) 2 (d) 9
  19. 19.Which of these numbers is divisible by 4? (a) 1017 (b) 1011 (c) 1012 (d) 1013
  20. 20.What is the remainder when 115 is divided by 6? (a) 2 (b) 1 (c) 5 (d) 0
  21. 21.What is the units digit of 3^14? (a) 7 (b) 9 (c) 1 (d) 3
  22. 22.What is the remainder when 459 is divided by 11? (a) 9 (b) 8 (c) 7 (d) 3
  23. 23.What is the remainder when 213 is divided by 12? (a) 9 (b) 10 (c) 3 (d) 8
  24. 24.What is the smallest number you must add to 113 to make it divisible by 11? (a) 8 (b) 7 (c) 9 (d) 3
  25. 25.Which of these numbers is divisible by 8? (a) 6047 (b) 6033 (c) 6032 (d) 6028
  26. 26.What is the remainder when 347 is divided by 10? (a) 3 (b) 8 (c) 7 (d) 6
  27. 27.What is the smallest number you must add to 53 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  28. 28.What is the smallest number you must add to 104 to make it divisible by 3? (a) 1 (b) 2
  29. 29.What is the smallest number you must add to 204 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
  30. 30.What is the units digit of 7^23? (a) 1 (b) 9 (c) 7 (d) 3
  31. 31.Which of these numbers is divisible by 9? (a) 3543 (b) 3547 (c) 3546 (d) 3557
  32. 32.What is the smallest number you must add to 585 to make it divisible by 8? (a) 5 (b) 1 (c) 7 (d) 6
  33. 33.Which of these numbers is divisible by 3? (a) 1041 (b) 1046 (c) 1039 (d) 1043
  34. 34.What is the remainder when 446 is divided by 12? (a) 10 (b) 1 (c) 3 (d) 2
  35. 35.What is the smallest number you must add to 122 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  36. 36.Which of these numbers is divisible by 11? (a) 1629 (b) 1644 (c) 1655 (d) 1639
  37. 37.What is the remainder when 510 is divided by 7? (a) 5 (b) 4 (c) 1 (d) 6
  38. 38.What is the remainder when 173 is divided by 3? (a) 0 (b) 2 (c) 1
  39. 39.What is the remainder when 261 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
  40. 40.What is the smallest number you must add to 217 to make it divisible by 4? (a) 1 (b) 2 (c) 3

Answer key

  1. 1392
  2. 4
  3. 10
  4. 432
  5. 1
  6. 5481
  7. 4
  8. 2
  9. 5
  10. 4
  11. 1917
  12. 9
  13. 2421
  14. 3536
  15. 5076
  16. 8
  17. 1756
  18. 2
  19. 1012
  20. 1
  21. 9
  22. 8
  23. 9
  24. 8
  25. 6032
  26. 7
  27. 3
  28. 1
  29. 1
  30. 3
  31. 3546
  32. 7
  33. 1041
  34. 2
  35. 2
  36. 1639
  37. 6
  38. 2
  39. 1
  40. 3

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