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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 782 · 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 406 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
  2. 2.Which of these numbers is divisible by 8? (a) 751 (b) 766 (c) 752 (d) 755
  3. 3.What is the smallest number you must add to 435 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 7
  4. 4.Which of these numbers is divisible by 6? (a) 3734 (b) 3740 (c) 3747 (d) 3738
  5. 5.What is the smallest number you must add to 285 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
  6. 6.What is the units digit of 7^22? (a) 7 (b) 9 (c) 1 (d) 3
  7. 7.What is the smallest number you must add to 233 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  8. 8.What is the smallest number you must add to 199 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
  9. 9.What is the smallest number you must add to 253 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  10. 10.Which of these numbers is divisible by 6? (a) 2689 (b) 2697 (c) 2688 (d) 2683
  11. 11.What is the units digit of 2^22? (a) 6 (b) 2 (c) 4 (d) 8
  12. 12.What is the smallest number you must add to 374 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
  13. 13.Which of these numbers is divisible by 3? (a) 169 (b) 167 (c) 164 (d) 165
  14. 14.What is the smallest number you must add to 496 to make it divisible by 9? (a) 7 (b) 8 (c) 1 (d) 6
  15. 15.What is the remainder when 136 is divided by 11? (a) 7 (b) 4 (c) 3 (d) 5
  16. 16.What is the remainder when 564 is divided by 11? (a) 2 (b) 8 (c) 3 (d) 4
  17. 17.What is the smallest number you must add to 679 to make it divisible by 10? (a) 3 (b) 2 (c) 9 (d) 1
  18. 18.What is the units digit of 7^47? (a) 7 (b) 3 (c) 9 (d) 1
  19. 19.What is the remainder when 241 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
  20. 20.What is the units digit of 9^23? (a) 7 (b) 9 (c) 8 (d) 1
  21. 21.Which of these numbers is divisible by 8? (a) 6834 (b) 6846 (c) 6827 (d) 6832
  22. 22.What is the remainder when 47 is divided by 3? (a) 2 (b) 0 (c) 1
  23. 23.What is the remainder when 50 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
  24. 24.What is the smallest number you must add to 232 to make it divisible by 3? (a) 1 (b) 2
  25. 25.Which of these numbers is divisible by 3? (a) 2395 (b) 2401 (c) 2397 (d) 2399
  26. 26.What is the smallest number you must add to 304 to make it divisible by 9? (a) 1 (b) 7 (c) 2 (d) 3
  27. 27.What is the units digit of 2^42? (a) 6 (b) 8 (c) 4 (d) 2
  28. 28.What is the remainder when 194 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
  29. 29.What is the smallest number you must add to 333 to make it divisible by 12? (a) 9 (b) 2 (c) 3 (d) 4
  30. 30.What is the remainder when 546 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 3
  31. 31.What is the smallest number you must add to 172 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 2
  32. 32.Which of these numbers is divisible by 3? (a) 2039 (b) 2033 (c) 2034 (d) 2036
  33. 33.What is the units digit of 9^22? (a) 2 (b) 9 (c) 0 (d) 1
  34. 34.Which of these numbers is divisible by 9? (a) 4050 (b) 4066 (c) 4057 (d) 4043
  35. 35.What is the units digit of 7^54? (a) 7 (b) 1 (c) 3 (d) 9
  36. 36.What is the remainder when 141 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  37. 37.What is the units digit of 4^19? (a) 3 (b) 5 (c) 6 (d) 4
  38. 38.What is the units digit of 7^10? (a) 7 (b) 9 (c) 1 (d) 3
  39. 39.What is the units digit of 9^26? (a) 9 (b) 0 (c) 1 (d) 2
  40. 40.What is the units digit of 7^25? (a) 9 (b) 7 (c) 3 (d) 1

Answer key

  1. 4
  2. 752
  3. 6
  4. 3738
  5. 3
  6. 9
  7. 3
  8. 1
  9. 3
  10. 2688
  11. 4
  12. 10
  13. 165
  14. 8
  15. 4
  16. 3
  17. 1
  18. 3
  19. 1
  20. 9
  21. 6832
  22. 2
  23. 2
  24. 2
  25. 2397
  26. 2
  27. 4
  28. 2
  29. 3
  30. 6
  31. 8
  32. 2034
  33. 1
  34. 4050
  35. 9
  36. 1
  37. 4
  38. 9
  39. 1
  40. 7

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