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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 910 · 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 2^19? (a) 8 (b) 2 (c) 6 (d) 4
  2. 2.What is the remainder when 664 is divided by 9? (a) 2 (b) 6 (c) 8 (d) 7
  3. 3.What is the smallest number you must add to 175 to make it divisible by 3? (a) 2 (b) 1
  4. 4.What is the remainder when 205 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
  5. 5.What is the smallest number you must add to 433 to make it divisible by 9? (a) 8 (b) 1 (c) 6 (d) 7
  6. 6.Which of these numbers is divisible by 9? (a) 6817 (b) 6811 (c) 6813 (d) 6829
  7. 7.What is the units digit of 7^47? (a) 3 (b) 9 (c) 1 (d) 7
  8. 8.What is the remainder when 123 is divided by 8? (a) 3 (b) 4 (c) 2 (d) 5
  9. 9.What is the remainder when 316 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
  10. 10.Which of these numbers is divisible by 6? (a) 5133 (b) 5137 (c) 5130 (d) 5125
  11. 11.What is the smallest number you must add to 67 to make it divisible by 3? (a) 1 (b) 2
  12. 12.What is the remainder when 127 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  13. 13.What is the remainder when 703 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
  14. 14.What is the smallest number you must add to 211 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  15. 15.What is the units digit of 4^42? (a) 6 (b) 4 (c) 5 (d) 7
  16. 16.What is the units digit of 9^14? (a) 2 (b) 9 (c) 1 (d) 0
  17. 17.What is the remainder when 673 is divided by 10? (a) 7 (b) 2 (c) 3 (d) 4
  18. 18.What is the smallest number you must add to 625 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
  19. 19.What is the remainder when 370 is divided by 7? (a) 4 (b) 1 (c) 5 (d) 6
  20. 20.What is the smallest number you must add to 169 to make it divisible by 12? (a) 10 (b) 11 (c) 1 (d) 9
  21. 21.What is the smallest number you must add to 122 to make it divisible by 7? (a) 4 (b) 6 (c) 3 (d) 5
  22. 22.What is the remainder when 164 is divided by 6? (a) 1 (b) 3 (c) 2 (d) 4
  23. 23.What is the remainder when 71 is divided by 3? (a) 1 (b) 0 (c) 2
  24. 24.What is the smallest number you must add to 293 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 2
  25. 25.What is the smallest number you must add to 309 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  26. 26.What is the remainder when 94 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
  27. 27.What is the smallest number you must add to 246 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  28. 28.What is the smallest number you must add to 89 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  29. 29.What is the units digit of 8^58? (a) 8 (b) 6 (c) 2 (d) 4
  30. 30.Which of these numbers is divisible by 3? (a) 827 (b) 830 (c) 825 (d) 823
  31. 31.What is the smallest number you must add to 273 to make it divisible by 10? (a) 8 (b) 6 (c) 3 (d) 7
  32. 32.Which of these numbers is divisible by 6? (a) 3982 (b) 3986 (c) 3995 (d) 3984
  33. 33.What is the units digit of 4^22? (a) 6 (b) 5 (c) 4 (d) 7
  34. 34.What is the smallest number you must add to 83 to make it divisible by 7? (a) 2 (b) 6 (c) 1 (d) 3
  35. 35.What is the remainder when 167 is divided by 3? (a) 1 (b) 0 (c) 2
  36. 36.What is the smallest number you must add to 61 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  37. 37.What is the units digit of 9^22? (a) 2 (b) 9 (c) 1 (d) 0
  38. 38.What is the smallest number you must add to 611 to make it divisible by 12? (a) 11 (b) 2 (c) 1 (d) 3
  39. 39.What is the smallest number you must add to 276 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 1
  40. 40.What is the units digit of 2^11? (a) 4 (b) 8 (c) 2 (d) 6

Answer key

  1. 8
  2. 7
  3. 2
  4. 1
  5. 8
  6. 6813
  7. 3
  8. 3
  9. 4
  10. 5130
  11. 2
  12. 3
  13. 3
  14. 4
  15. 6
  16. 1
  17. 3
  18. 7
  19. 6
  20. 11
  21. 4
  22. 2
  23. 2
  24. 3
  25. 3
  26. 4
  27. 2
  28. 1
  29. 4
  30. 825
  31. 7
  32. 3984
  33. 6
  34. 1
  35. 2
  36. 4
  37. 1
  38. 1
  39. 10
  40. 8

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