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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 711 · 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) 2162 (b) 2167 (c) 2160 (d) 2156
  2. 2.What is the units digit of 4^42? (a) 6 (b) 7 (c) 4 (d) 5
  3. 3.Which of these numbers is divisible by 6? (a) 1026 (b) 1034 (c) 1030 (d) 1023
  4. 4.What is the remainder when 58 is divided by 3? (a) 1 (b) 2 (c) 0
  5. 5.Which of these numbers is divisible by 6? (a) 4875 (b) 4889 (c) 4878 (d) 4881
  6. 6.What is the units digit of 2^50? (a) 6 (b) 8 (c) 2 (d) 4
  7. 7.What is the units digit of 8^52? (a) 4 (b) 2 (c) 6 (d) 8
  8. 8.Which of these numbers is divisible by 3? (a) 694 (b) 690 (c) 692 (d) 688
  9. 9.What is the units digit of 8^27? (a) 2 (b) 4 (c) 8 (d) 6
  10. 10.What is the remainder when 631 is divided by 8? (a) 6 (b) 1 (c) 5 (d) 7
  11. 11.What is the remainder when 679 is divided by 11? (a) 3 (b) 9 (c) 7 (d) 8
  12. 12.What is the smallest number you must add to 297 to make it divisible by 7? (a) 6 (b) 4 (c) 3 (d) 5
  13. 13.What is the remainder when 403 is divided by 10? (a) 3 (b) 7 (c) 4 (d) 2
  14. 14.Which of these numbers is divisible by 4? (a) 2597 (b) 2600 (c) 2601 (d) 2607
  15. 15.What is the remainder when 306 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
  16. 16.Which of these numbers is divisible by 11? (a) 1227 (b) 1210 (c) 1209 (d) 1220
  17. 17.What is the remainder when 74 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 6
  18. 18.What is the smallest number you must add to 95 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3
  19. 19.What is the remainder when 559 is divided by 9? (a) 8 (b) 0 (c) 1 (d) 2
  20. 20.What is the smallest number you must add to 67 to make it divisible by 3? (a) 2 (b) 1
  21. 21.What is the remainder when 246 is divided by 8? (a) 5 (b) 6 (c) 7 (d) 2
  22. 22.Which of these numbers is divisible by 6? (a) 5374 (b) 5369 (c) 5378 (d) 5370
  23. 23.What is the units digit of 7^46? (a) 1 (b) 7 (c) 3 (d) 9
  24. 24.What is the remainder when 259 is divided by 12? (a) 7 (b) 6 (c) 5 (d) 8
  25. 25.What is the smallest number you must add to 141 to make it divisible by 11? (a) 2 (b) 9 (c) 1 (d) 3
  26. 26.What is the smallest number you must add to 484 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
  27. 27.What is the smallest number you must add to 400 to make it divisible by 11? (a) 6 (b) 8 (c) 7 (d) 4
  28. 28.What is the smallest number you must add to 433 to make it divisible by 10? (a) 6 (b) 8 (c) 7 (d) 3
  29. 29.What is the smallest number you must add to 43 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  30. 30.What is the remainder when 357 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5
  31. 31.Which of these numbers is divisible by 6? (a) 1109 (b) 1111 (c) 1110 (d) 1117
  32. 32.What is the smallest number you must add to 531 to make it divisible by 12? (a) 8 (b) 10 (c) 3 (d) 9
  33. 33.What is the remainder when 800 is divided by 11? (a) 9 (b) 7 (c) 3 (d) 8
  34. 34.Which of these numbers is divisible by 4? (a) 3117 (b) 3126 (c) 3122 (d) 3120
  35. 35.What is the remainder when 238 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
  36. 36.What is the smallest number you must add to 508 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
  37. 37.What is the remainder when 172 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
  38. 38.Which of these numbers is divisible by 6? (a) 2518 (b) 2522 (c) 2529 (d) 2520
  39. 39.What is the smallest number you must add to 943 to make it divisible by 12? (a) 6 (b) 4 (c) 7 (d) 5
  40. 40.What is the smallest number you must add to 193 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4

Answer key

  1. 2160
  2. 6
  3. 1026
  4. 1
  5. 4878
  6. 4
  7. 6
  8. 690
  9. 2
  10. 7
  11. 8
  12. 4
  13. 3
  14. 2600
  15. 6
  16. 1210
  17. 4
  18. 1
  19. 1
  20. 2
  21. 6
  22. 5370
  23. 9
  24. 7
  25. 2
  26. 2
  27. 7
  28. 7
  29. 1
  30. 3
  31. 1110
  32. 9
  33. 8
  34. 3120
  35. 4
  36. 4
  37. 2
  38. 2520
  39. 5
  40. 2

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