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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 59 · 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 9^45? (a) 9 (b) 7 (c) 8 (d) 1
  2. 2.Which of these numbers is divisible by 8? (a) 944 (b) 955 (c) 938 (d) 949
  3. 3.What is the smallest number you must add to 396 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
  4. 4.What is the remainder when 92 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
  5. 5.What is the remainder when 617 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
  6. 6.What is the remainder when 476 is divided by 11? (a) 4 (b) 3 (c) 8 (d) 2
  7. 7.What is the units digit of 7^12? (a) 7 (b) 9 (c) 3 (d) 1
  8. 8.What is the units digit of 4^46? (a) 4 (b) 5 (c) 6 (d) 7
  9. 9.What is the smallest number you must add to 43 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  10. 10.What is the remainder when 264 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
  11. 11.What is the units digit of 8^45? (a) 2 (b) 8 (c) 6 (d) 4
  12. 12.What is the units digit of 9^44? (a) 0 (b) 2 (c) 1 (d) 9
  13. 13.What is the smallest number you must add to 235 to make it divisible by 10? (a) 4 (b) 7 (c) 6 (d) 5
  14. 14.What is the units digit of 4^30? (a) 4 (b) 5 (c) 6 (d) 7
  15. 15.What is the remainder when 461 is divided by 9? (a) 2 (b) 1 (c) 7 (d) 3
  16. 16.What is the remainder when 808 is divided by 12? (a) 4 (b) 3 (c) 8 (d) 5
  17. 17.What is the units digit of 7^32? (a) 3 (b) 9 (c) 1 (d) 7
  18. 18.What is the smallest number you must add to 352 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
  19. 19.Which of these numbers is divisible by 11? (a) 6161 (b) 6157 (c) 6149 (d) 6148
  20. 20.What is the remainder when 254 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
  21. 21.What is the remainder when 348 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 3
  22. 22.What is the remainder when 170 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
  23. 23.What is the remainder when 97 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
  24. 24.What is the smallest number you must add to 752 to make it divisible by 10? (a) 8 (b) 2 (c) 9 (d) 7
  25. 25.What is the remainder when 638 is divided by 12? (a) 2 (b) 10 (c) 3 (d) 1
  26. 26.What is the smallest number you must add to 717 to make it divisible by 11? (a) 2 (b) 10 (c) 8 (d) 9
  27. 27.What is the units digit of 4^56? (a) 7 (b) 6 (c) 5 (d) 4
  28. 28.Which of these numbers is divisible by 6? (a) 5323 (b) 5337 (c) 5331 (d) 5328
  29. 29.What is the remainder when 640 is divided by 11? (a) 2 (b) 3 (c) 1 (d) 9
  30. 30.What is the units digit of 3^10? (a) 3 (b) 1 (c) 7 (d) 9
  31. 31.What is the remainder when 169 is divided by 6? (a) 5 (b) 1 (c) 0 (d) 2
  32. 32.Which of these numbers is divisible by 11? (a) 5824 (b) 5798 (c) 5811 (d) 5808
  33. 33.What is the smallest number you must add to 601 to make it divisible by 12? (a) 10 (b) 9 (c) 11 (d) 1
  34. 34.What is the smallest number you must add to 429 to make it divisible by 12? (a) 3 (b) 9 (c) 2 (d) 4
  35. 35.What is the smallest number you must add to 94 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  36. 36.Which of these numbers is divisible by 8? (a) 3630 (b) 3618 (c) 3639 (d) 3624
  37. 37.What is the remainder when 275 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 4
  38. 38.What is the smallest number you must add to 163 to make it divisible by 3? (a) 1 (b) 2
  39. 39.Which of these numbers is divisible by 4? (a) 1413 (b) 1410 (c) 1408 (d) 1405
  40. 40.What is the remainder when 442 is divided by 7? (a) 2 (b) 6 (c) 0 (d) 1

Answer key

  1. 9
  2. 944
  3. 3
  4. 2
  5. 1
  6. 3
  7. 1
  8. 6
  9. 1
  10. 4
  11. 8
  12. 1
  13. 5
  14. 6
  15. 2
  16. 4
  17. 1
  18. 2
  19. 6149
  20. 4
  21. 6
  22. 2
  23. 2
  24. 8
  25. 2
  26. 9
  27. 6
  28. 5328
  29. 2
  30. 9
  31. 1
  32. 5808
  33. 11
  34. 3
  35. 2
  36. 3624
  37. 5
  38. 2
  39. 1408
  40. 1

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