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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 96 · 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 remainder when 229 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
  2. 2.What is the remainder when 792 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
  3. 3.What is the units digit of 3^52? (a) 3 (b) 7 (c) 9 (d) 1
  4. 4.What is the units digit of 4^16? (a) 4 (b) 6 (c) 7 (d) 5
  5. 5.What is the smallest number you must add to 192 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 8
  6. 6.What is the remainder when 678 is divided by 10? (a) 2 (b) 7 (c) 9 (d) 8
  7. 7.What is the smallest number you must add to 698 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 5
  8. 8.What is the smallest number you must add to 100 to make it divisible by 3? (a) 2 (b) 1
  9. 9.Which of these numbers is divisible by 4? (a) 745 (b) 749 (c) 744 (d) 743
  10. 10.What is the smallest number you must add to 148 to make it divisible by 7? (a) 5 (b) 6 (c) 1 (d) 4
  11. 11.What is the smallest number you must add to 865 to make it divisible by 12? (a) 10 (b) 1 (c) 9 (d) 11
  12. 12.What is the units digit of 2^21? (a) 4 (b) 2 (c) 6 (d) 8
  13. 13.What is the smallest number you must add to 662 to make it divisible by 12? (a) 10 (b) 11 (c) 2 (d) 9
  14. 14.Which of these numbers is divisible by 6? (a) 3768 (b) 3777 (c) 3771 (d) 3766
  15. 15.What is the units digit of 8^21? (a) 4 (b) 8 (c) 2 (d) 6
  16. 16.Which of these numbers is divisible by 4? (a) 2514 (b) 2518 (c) 2512 (d) 2510
  17. 17.What is the remainder when 719 is divided by 9? (a) 7 (b) 6 (c) 8 (d) 1
  18. 18.What is the remainder when 547 is divided by 9? (a) 2 (b) 8 (c) 7 (d) 6
  19. 19.What is the remainder when 405 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
  20. 20.What is the units digit of 2^53? (a) 2 (b) 8 (c) 6 (d) 4
  21. 21.What is the smallest number you must add to 133 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 4
  22. 22.What is the smallest number you must add to 109 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3
  23. 23.What is the remainder when 861 is divided by 11? (a) 3 (b) 8 (c) 2 (d) 4
  24. 24.What is the remainder when 353 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
  25. 25.Which of these numbers is divisible by 9? (a) 1961 (b) 1946 (c) 1963 (d) 1953
  26. 26.What is the units digit of 8^22? (a) 2 (b) 4 (c) 6 (d) 8
  27. 27.What is the smallest number you must add to 242 to make it divisible by 3? (a) 2 (b) 1
  28. 28.What is the remainder when 521 is divided by 7? (a) 4 (b) 2 (c) 3 (d) 5
  29. 29.What is the smallest number you must add to 196 to make it divisible by 9? (a) 7 (b) 1 (c) 2 (d) 3
  30. 30.What is the smallest number you must add to 211 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
  31. 31.What is the smallest number you must add to 964 to make it divisible by 12? (a) 8 (b) 4 (c) 9 (d) 7
  32. 32.What is the smallest number you must add to 439 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
  33. 33.Which of these numbers is divisible by 9? (a) 2711 (b) 2702 (c) 2722 (d) 2709
  34. 34.What is the smallest number you must add to 586 to make it divisible by 12? (a) 3 (b) 10 (c) 1 (d) 2
  35. 35.What is the smallest number you must add to 753 to make it divisible by 12? (a) 9 (b) 3 (c) 4 (d) 2
  36. 36.What is the remainder when 191 is divided by 7? (a) 3 (b) 1 (c) 2 (d) 5
  37. 37.What is the units digit of 4^11? (a) 6 (b) 3 (c) 4 (d) 5
  38. 38.Which of these numbers is divisible by 6? (a) 4140 (b) 4141 (c) 4136 (d) 4150
  39. 39.What is the remainder when 642 is divided by 8? (a) 6 (b) 1 (c) 3 (d) 2
  40. 40.What is the smallest number you must add to 257 to make it divisible by 4? (a) 2 (b) 1 (c) 3

Answer key

  1. 1
  2. 2
  3. 1
  4. 6
  5. 6
  6. 8
  7. 6
  8. 2
  9. 744
  10. 6
  11. 11
  12. 2
  13. 10
  14. 3768
  15. 8
  16. 2512
  17. 8
  18. 7
  19. 5
  20. 2
  21. 5
  22. 3
  23. 3
  24. 5
  25. 1953
  26. 4
  27. 1
  28. 3
  29. 2
  30. 5
  31. 8
  32. 5
  33. 2709
  34. 2
  35. 3
  36. 2
  37. 4
  38. 4140
  39. 2
  40. 3

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