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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 933 · 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 100 is divided by 7? (a) 1 (b) 5 (c) 2 (d) 3
  2. 2.What is the remainder when 247 is divided by 12? (a) 8 (b) 6 (c) 5 (d) 7
  3. 3.What is the units digit of 8^30? (a) 8 (b) 4 (c) 6 (d) 2
  4. 4.Which of these numbers is divisible by 11? (a) 4063 (b) 4056 (c) 4048 (d) 4044
  5. 5.What is the remainder when 249 is divided by 10? (a) 1 (b) 8 (c) 9 (d) 7
  6. 6.Which of these numbers is divisible by 8? (a) 5000 (b) 4998 (c) 5015 (d) 5003
  7. 7.What is the remainder when 242 is divided by 3? (a) 1 (b) 0 (c) 2
  8. 8.Which of these numbers is divisible by 3? (a) 854 (b) 848 (c) 849 (d) 850
  9. 9.What is the smallest number you must add to 292 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  10. 10.What is the smallest number you must add to 259 to make it divisible by 10? (a) 9 (b) 3 (c) 1 (d) 2
  11. 11.What is the units digit of 9^43? (a) 8 (b) 9 (c) 7 (d) 1
  12. 12.What is the smallest number you must add to 383 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  13. 13.What is the remainder when 305 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
  14. 14.What is the remainder when 218 is divided by 3? (a) 1 (b) 2 (c) 0
  15. 15.What is the smallest number you must add to 94 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 6
  16. 16.What is the smallest number you must add to 535 to make it divisible by 7? (a) 4 (b) 3 (c) 6 (d) 5
  17. 17.What is the smallest number you must add to 131 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  18. 18.Which of these numbers is divisible by 11? (a) 4301 (b) 4314 (c) 4296 (d) 4310
  19. 19.Which of these numbers is divisible by 3? (a) 1895 (b) 1900 (c) 1896 (d) 1898
  20. 20.What is the remainder when 181 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  21. 21.What is the units digit of 9^47? (a) 7 (b) 9 (c) 1 (d) 8
  22. 22.What is the units digit of 8^35? (a) 4 (b) 6 (c) 8 (d) 2
  23. 23.What is the smallest number you must add to 95 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  24. 24.Which of these numbers is divisible by 4? (a) 856 (b) 855 (c) 862 (d) 858
  25. 25.What is the remainder when 244 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
  26. 26.What is the smallest number you must add to 119 to make it divisible by 3? (a) 2 (b) 1
  27. 27.What is the remainder when 213 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
  28. 28.What is the smallest number you must add to 137 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  29. 29.What is the units digit of 9^31? (a) 1 (b) 8 (c) 7 (d) 9
  30. 30.What is the remainder when 363 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
  31. 31.What is the remainder when 638 is divided by 9? (a) 1 (b) 6 (c) 8 (d) 7
  32. 32.What is the remainder when 484 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 3
  33. 33.What is the smallest number you must add to 479 to make it divisible by 9? (a) 8 (b) 6 (c) 2 (d) 7
  34. 34.Which of these numbers is divisible by 9? (a) 6807 (b) 6795 (c) 6794 (d) 6797
  35. 35.Which of these numbers is divisible by 8? (a) 1677 (b) 1685 (c) 1672 (d) 1667
  36. 36.What is the remainder when 116 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
  37. 37.What is the remainder when 310 is divided by 9? (a) 6 (b) 3 (c) 4 (d) 5
  38. 38.What is the remainder when 161 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  39. 39.What is the units digit of 3^57? (a) 7 (b) 9 (c) 3 (d) 1
  40. 40.Which of these numbers is divisible by 11? (a) 7258 (b) 7278 (c) 7270 (d) 7260

Answer key

  1. 2
  2. 7
  3. 4
  4. 4048
  5. 9
  6. 5000
  7. 2
  8. 849
  9. 3
  10. 1
  11. 9
  12. 2
  13. 1
  14. 2
  15. 4
  16. 4
  17. 1
  18. 4301
  19. 1896
  20. 1
  21. 9
  22. 2
  23. 1
  24. 856
  25. 4
  26. 1
  27. 5
  28. 3
  29. 9
  30. 3
  31. 8
  32. 4
  33. 7
  34. 6795
  35. 1672
  36. 6
  37. 4
  38. 1
  39. 3
  40. 7260

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