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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 227 · 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 4? (a) 727 (b) 724 (c) 730 (d) 723
  2. 2.What is the units digit of 7^42? (a) 3 (b) 7 (c) 1 (d) 9
  3. 3.What is the smallest number you must add to 444 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
  4. 4.What is the remainder when 354 is divided by 11? (a) 3 (b) 2 (c) 1 (d) 9
  5. 5.What is the remainder when 201 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 2
  6. 6.Which of these numbers is divisible by 11? (a) 767 (b) 780 (c) 759 (d) 752
  7. 7.Which of these numbers is divisible by 9? (a) 8071 (b) 8055 (c) 8053 (d) 8060
  8. 8.What is the smallest number you must add to 486 to make it divisible by 10? (a) 4 (b) 6 (c) 3 (d) 5
  9. 9.What is the smallest number you must add to 62 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 5
  10. 10.What is the units digit of 3^48? (a) 7 (b) 9 (c) 3 (d) 1
  11. 11.What is the smallest number you must add to 632 to make it divisible by 12? (a) 3 (b) 4 (c) 5 (d) 8
  12. 12.What is the remainder when 106 is divided by 3? (a) 0 (b) 2 (c) 1
  13. 13.Which of these numbers is divisible by 4? (a) 2772 (b) 2777 (c) 2775 (d) 2770
  14. 14.What is the units digit of 9^44? (a) 1 (b) 0 (c) 2 (d) 9
  15. 15.What is the remainder when 322 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
  16. 16.What is the units digit of 2^31? (a) 4 (b) 6 (c) 2 (d) 8
  17. 17.What is the remainder when 233 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
  18. 18.What is the smallest number you must add to 913 to make it divisible by 12? (a) 1 (b) 9 (c) 11 (d) 10
  19. 19.What is the smallest number you must add to 154 to make it divisible by 3? (a) 1 (b) 2
  20. 20.What is the remainder when 417 is divided by 8? (a) 7 (b) 1 (c) 0 (d) 2
  21. 21.What is the units digit of 9^27? (a) 8 (b) 7 (c) 9 (d) 1
  22. 22.What is the remainder when 504 is divided by 11? (a) 8 (b) 2 (c) 9 (d) 10
  23. 23.What is the units digit of 3^40? (a) 7 (b) 1 (c) 9 (d) 3
  24. 24.What is the smallest number you must add to 460 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
  25. 25.What is the units digit of 3^10? (a) 7 (b) 3 (c) 9 (d) 1
  26. 26.What is the remainder when 280 is divided by 12? (a) 8 (b) 4 (c) 5 (d) 3
  27. 27.What is the smallest number you must add to 391 to make it divisible by 6? (a) 5 (b) 1 (c) 4 (d) 3
  28. 28.What is the units digit of 3^18? (a) 1 (b) 3 (c) 9 (d) 7
  29. 29.What is the remainder when 192 is divided by 10? (a) 8 (b) 2 (c) 3 (d) 1
  30. 30.What is the smallest number you must add to 199 to make it divisible by 9? (a) 1 (b) 8 (c) 7 (d) 6
  31. 31.What is the units digit of 4^38? (a) 4 (b) 6 (c) 7 (d) 5
  32. 32.What is the units digit of 2^13? (a) 6 (b) 8 (c) 4 (d) 2
  33. 33.What is the units digit of 3^24? (a) 7 (b) 1 (c) 9 (d) 3
  34. 34.What is the smallest number you must add to 499 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
  35. 35.What is the smallest number you must add to 99 to make it divisible by 7? (a) 6 (b) 5 (c) 1 (d) 4
  36. 36.What is the smallest number you must add to 540 to make it divisible by 11? (a) 8 (b) 9 (c) 10 (d) 1
  37. 37.What is the remainder when 361 is divided by 11? (a) 9 (b) 10 (c) 8 (d) 2
  38. 38.What is the smallest number you must add to 313 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  39. 39.What is the units digit of 2^35? (a) 8 (b) 2 (c) 6 (d) 4
  40. 40.Which of these numbers is divisible by 6? (a) 4707 (b) 4698 (c) 4695 (d) 4699

Answer key

  1. 724
  2. 9
  3. 4
  4. 2
  5. 3
  6. 759
  7. 8055
  8. 4
  9. 4
  10. 1
  11. 4
  12. 1
  13. 2772
  14. 1
  15. 2
  16. 8
  17. 2
  18. 11
  19. 2
  20. 1
  21. 9
  22. 9
  23. 1
  24. 8
  25. 9
  26. 4
  27. 5
  28. 9
  29. 2
  30. 8
  31. 6
  32. 2
  33. 1
  34. 5
  35. 6
  36. 10
  37. 9
  38. 3
  39. 8
  40. 4698

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