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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 764 · 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 261 is divided by 11? (a) 9 (b) 8 (c) 3 (d) 7
  2. 2.What is the smallest number you must add to 188 to make it divisible by 3? (a) 1 (b) 2
  3. 3.What is the remainder when 321 is divided by 10? (a) 2 (b) 0 (c) 1 (d) 9
  4. 4.What is the units digit of 3^31? (a) 3 (b) 7 (c) 1 (d) 9
  5. 5.Which of these numbers is divisible by 11? (a) 7150 (b) 7143 (c) 7168 (d) 7153
  6. 6.What is the remainder when 105 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  7. 7.Which of these numbers is divisible by 3? (a) 2138 (b) 2139 (c) 2140 (d) 2144
  8. 8.Which of these numbers is divisible by 4? (a) 2649 (b) 2653 (c) 2657 (d) 2652
  9. 9.What is the smallest number you must add to 129 to make it divisible by 8? (a) 7 (b) 5 (c) 1 (d) 6
  10. 10.Which of these numbers is divisible by 6? (a) 4218 (b) 4215 (c) 4229 (d) 4220
  11. 11.What is the units digit of 8^29? (a) 8 (b) 2 (c) 6 (d) 4
  12. 12.What is the smallest number you must add to 210 to make it divisible by 11? (a) 8 (b) 9 (c) 1 (d) 10
  13. 13.What is the smallest number you must add to 68 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  14. 14.What is the remainder when 174 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
  15. 15.What is the smallest number you must add to 195 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 6
  16. 16.What is the units digit of 2^18? (a) 8 (b) 4 (c) 6 (d) 2
  17. 17.Which of these numbers is divisible by 9? (a) 4401 (b) 4395 (c) 4409 (d) 4415
  18. 18.Which of these numbers is divisible by 9? (a) 3611 (b) 3600 (c) 3601 (d) 3599
  19. 19.What is the smallest number you must add to 148 to make it divisible by 3? (a) 1 (b) 2
  20. 20.What is the units digit of 3^43? (a) 9 (b) 7 (c) 3 (d) 1
  21. 21.What is the smallest number you must add to 196 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  22. 22.Which of these numbers is divisible by 11? (a) 6160 (b) 6159 (c) 6161 (d) 6173
  23. 23.What is the remainder when 718 is divided by 9? (a) 2 (b) 7 (c) 6 (d) 8
  24. 24.Which of these numbers is divisible by 3? (a) 2664 (b) 2665 (c) 2668 (d) 2663
  25. 25.What is the units digit of 4^15? (a) 6 (b) 5 (c) 4 (d) 3
  26. 26.What is the units digit of 4^45? (a) 5 (b) 6 (c) 4 (d) 3
  27. 27.What is the units digit of 7^38? (a) 3 (b) 7 (c) 9 (d) 1
  28. 28.What is the smallest number you must add to 115 to make it divisible by 11? (a) 7 (b) 5 (c) 8 (d) 6
  29. 29.What is the smallest number you must add to 87 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 3
  30. 30.What is the units digit of 3^33? (a) 7 (b) 1 (c) 9 (d) 3
  31. 31.What is the units digit of 9^27? (a) 1 (b) 8 (c) 9 (d) 7
  32. 32.What is the remainder when 163 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
  33. 33.What is the smallest number you must add to 137 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  34. 34.What is the remainder when 271 is divided by 11? (a) 8 (b) 6 (c) 4 (d) 7
  35. 35.What is the smallest number you must add to 634 to make it divisible by 8? (a) 2 (b) 6 (c) 5 (d) 7
  36. 36.Which of these numbers is divisible by 4? (a) 1029 (b) 1027 (c) 1034 (d) 1028
  37. 37.What is the remainder when 149 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
  38. 38.What is the remainder when 164 is divided by 3? (a) 1 (b) 0 (c) 2
  39. 39.What is the smallest number you must add to 456 to make it divisible by 9? (a) 2 (b) 4 (c) 6 (d) 3
  40. 40.What is the units digit of 9^53? (a) 1 (b) 9 (c) 8 (d) 7

Answer key

  1. 8
  2. 1
  3. 1
  4. 7
  5. 7150
  6. 1
  7. 2139
  8. 2652
  9. 7
  10. 4218
  11. 8
  12. 10
  13. 2
  14. 2
  15. 1
  16. 4
  17. 4401
  18. 3600
  19. 2
  20. 7
  21. 4
  22. 6160
  23. 7
  24. 2664
  25. 4
  26. 4
  27. 9
  28. 6
  29. 4
  30. 3
  31. 9
  32. 1
  33. 3
  34. 7
  35. 6
  36. 1028
  37. 1
  38. 2
  39. 3
  40. 9

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