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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 356 · 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 3? (a) 1107 (b) 1105 (c) 1111 (d) 1108
  2. 2.What is the units digit of 3^34? (a) 9 (b) 1 (c) 7 (d) 3
  3. 3.Which of these numbers is divisible by 11? (a) 3910 (b) 3921 (c) 3900 (d) 3905
  4. 4.What is the smallest number you must add to 567 to make it divisible by 10? (a) 2 (b) 3 (c) 4 (d) 7
  5. 5.What is the remainder when 332 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
  6. 6.What is the remainder when 772 is divided by 12? (a) 8 (b) 5 (c) 4 (d) 3
  7. 7.What is the units digit of 3^10? (a) 9 (b) 7 (c) 3 (d) 1
  8. 8.What is the units digit of 7^42? (a) 7 (b) 9 (c) 3 (d) 1
  9. 9.What is the units digit of 2^16? (a) 8 (b) 2 (c) 6 (d) 4
  10. 10.What is the units digit of 4^47? (a) 6 (b) 4 (c) 5 (d) 3
  11. 11.What is the units digit of 7^17? (a) 1 (b) 3 (c) 7 (d) 9
  12. 12.What is the smallest number you must add to 303 to make it divisible by 9? (a) 3 (b) 6 (c) 4 (d) 2
  13. 13.What is the smallest number you must add to 298 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
  14. 14.What is the remainder when 214 is divided by 7? (a) 3 (b) 6 (c) 4 (d) 5
  15. 15.What is the remainder when 492 is divided by 7? (a) 3 (b) 1 (c) 2 (d) 5
  16. 16.What is the smallest number you must add to 56 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  17. 17.What is the remainder when 379 is divided by 10? (a) 9 (b) 7 (c) 1 (d) 8
  18. 18.What is the remainder when 147 is divided by 10? (a) 8 (b) 3 (c) 6 (d) 7
  19. 19.What is the smallest number you must add to 295 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
  20. 20.Which of these numbers is divisible by 4? (a) 3400 (b) 3406 (c) 3399 (d) 3403
  21. 21.What is the remainder when 255 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
  22. 22.What is the units digit of 9^32? (a) 1 (b) 9 (c) 0 (d) 2
  23. 23.What is the smallest number you must add to 175 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  24. 24.What is the remainder when 317 is divided by 12? (a) 6 (b) 7 (c) 4 (d) 5
  25. 25.What is the units digit of 9^47? (a) 7 (b) 9 (c) 8 (d) 1
  26. 26.What is the remainder when 158 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
  27. 27.What is the smallest number you must add to 484 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 7
  28. 28.What is the remainder when 193 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  29. 29.What is the smallest number you must add to 632 to make it divisible by 9? (a) 2 (b) 8 (c) 7 (d) 6
  30. 30.What is the smallest number you must add to 241 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
  31. 31.Which of these numbers is divisible by 11? (a) 888 (b) 866 (c) 869 (d) 872
  32. 32.Which of these numbers is divisible by 4? (a) 424 (b) 426 (c) 423 (d) 430
  33. 33.What is the smallest number you must add to 707 to make it divisible by 9? (a) 6 (b) 5 (c) 3 (d) 4
  34. 34.What is the units digit of 7^21? (a) 3 (b) 9 (c) 1 (d) 7
  35. 35.What is the remainder when 175 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
  36. 36.What is the remainder when 62 is divided by 3? (a) 2 (b) 1 (c) 0
  37. 37.What is the smallest number you must add to 755 to make it divisible by 12? (a) 1 (b) 11 (c) 2 (d) 3
  38. 38.What is the smallest number you must add to 146 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  39. 39.What is the smallest number you must add to 158 to make it divisible by 9? (a) 6 (b) 5 (c) 4 (d) 3
  40. 40.What is the smallest number you must add to 110 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 6

Answer key

  1. 1107
  2. 9
  3. 3905
  4. 3
  5. 4
  6. 4
  7. 9
  8. 9
  9. 6
  10. 4
  11. 7
  12. 3
  13. 2
  14. 4
  15. 2
  16. 4
  17. 9
  18. 7
  19. 6
  20. 3400
  21. 3
  22. 1
  23. 1
  24. 5
  25. 9
  26. 3
  27. 6
  28. 3
  29. 7
  30. 4
  31. 869
  32. 424
  33. 4
  34. 7
  35. 3
  36. 2
  37. 1
  38. 2
  39. 4
  40. 2

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