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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 843 · 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 203 is divided by 10? (a) 7 (b) 3 (c) 2 (d) 4
  2. 2.What is the units digit of 3^12? (a) 1 (b) 3 (c) 7 (d) 9
  3. 3.What is the smallest number you must add to 311 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 1
  4. 4.What is the units digit of 7^11? (a) 3 (b) 7 (c) 9 (d) 1
  5. 5.What is the smallest number you must add to 335 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
  6. 6.Which of these numbers is divisible by 8? (a) 4131 (b) 4146 (c) 4136 (d) 4139
  7. 7.Which of these numbers is divisible by 3? (a) 1291 (b) 1294 (c) 1298 (d) 1293
  8. 8.Which of these numbers is divisible by 11? (a) 4323 (b) 4318 (c) 4328 (d) 4341
  9. 9.Which of these numbers is divisible by 6? (a) 736 (b) 730 (c) 721 (d) 726
  10. 10.What is the remainder when 122 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
  11. 11.Which of these numbers is divisible by 9? (a) 1234 (b) 1220 (c) 1227 (d) 1224
  12. 12.Which of these numbers is divisible by 3? (a) 875 (b) 868 (c) 872 (d) 870
  13. 13.What is the smallest number you must add to 631 to make it divisible by 11? (a) 6 (b) 4 (c) 7 (d) 8
  14. 14.What is the smallest number you must add to 325 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
  15. 15.What is the remainder when 477 is divided by 11? (a) 4 (b) 7 (c) 5 (d) 3
  16. 16.What is the remainder when 475 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
  17. 17.What is the units digit of 8^34? (a) 4 (b) 2 (c) 8 (d) 6
  18. 18.What is the remainder when 194 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
  19. 19.What is the units digit of 7^35? (a) 3 (b) 1 (c) 9 (d) 7
  20. 20.What is the smallest number you must add to 344 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
  21. 21.What is the remainder when 627 is divided by 10? (a) 8 (b) 6 (c) 3 (d) 7
  22. 22.What is the smallest number you must add to 123 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
  23. 23.What is the smallest number you must add to 98 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  24. 24.Which of these numbers is divisible by 3? (a) 658 (b) 662 (c) 656 (d) 657
  25. 25.What is the smallest number you must add to 664 to make it divisible by 9? (a) 1 (b) 7 (c) 3 (d) 2
  26. 26.What is the units digit of 7^33? (a) 3 (b) 7 (c) 9 (d) 1
  27. 27.Which of these numbers is divisible by 3? (a) 521 (b) 516 (c) 514 (d) 517
  28. 28.What is the smallest number you must add to 233 to make it divisible by 10? (a) 6 (b) 3 (c) 8 (d) 7
  29. 29.What is the units digit of 9^28? (a) 1 (b) 9 (c) 2 (d) 0
  30. 30.Which of these numbers is divisible by 8? (a) 4341 (b) 4359 (c) 4345 (d) 4344
  31. 31.What is the units digit of 2^39? (a) 4 (b) 8 (c) 6 (d) 2
  32. 32.What is the smallest number you must add to 109 to make it divisible by 3? (a) 1 (b) 2
  33. 33.What is the units digit of 4^31? (a) 3 (b) 4 (c) 5 (d) 6
  34. 34.Which of these numbers is divisible by 11? (a) 1936 (b) 1942 (c) 1935 (d) 1957
  35. 35.Which of these numbers is divisible by 8? (a) 779 (b) 784 (c) 793 (d) 786
  36. 36.What is the smallest number you must add to 171 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  37. 37.What is the smallest number you must add to 348 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 2
  38. 38.Which of these numbers is divisible by 11? (a) 7629 (b) 7612 (c) 7615 (d) 7606
  39. 39.Which of these numbers is divisible by 3? (a) 2633 (b) 2630 (c) 2635 (d) 2631
  40. 40.What is the smallest number you must add to 154 to make it divisible by 9? (a) 1 (b) 6 (c) 8 (d) 7

Answer key

  1. 3
  2. 1
  3. 1
  4. 3
  5. 5
  6. 4136
  7. 1293
  8. 4323
  9. 726
  10. 2
  11. 1224
  12. 870
  13. 7
  14. 8
  15. 4
  16. 1
  17. 4
  18. 4
  19. 3
  20. 4
  21. 7
  22. 3
  23. 2
  24. 657
  25. 2
  26. 7
  27. 516
  28. 7
  29. 1
  30. 4344
  31. 8
  32. 2
  33. 4
  34. 1936
  35. 784
  36. 1
  37. 3
  38. 7612
  39. 2631
  40. 8

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