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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 561 · 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) 703 (b) 701 (c) 702 (d) 707
  2. 2.What is the smallest number you must add to 299 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
  3. 3.Which of these numbers is divisible by 11? (a) 6991 (b) 6985 (c) 6998 (d) 6976
  4. 4.What is the units digit of 7^28? (a) 1 (b) 7 (c) 3 (d) 9
  5. 5.What is the remainder when 343 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
  6. 6.What is the smallest number you must add to 197 to make it divisible by 9? (a) 1 (b) 2 (c) 8 (d) 3
  7. 7.What is the remainder when 204 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 3
  8. 8.What is the smallest number you must add to 401 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  9. 9.Which of these numbers is divisible by 6? (a) 1062 (b) 1061 (c) 1073 (d) 1064
  10. 10.What is the units digit of 2^52? (a) 2 (b) 4 (c) 6 (d) 8
  11. 11.What is the smallest number you must add to 128 to make it divisible by 3? (a) 1 (b) 2
  12. 12.What is the units digit of 9^56? (a) 0 (b) 1 (c) 9 (d) 2
  13. 13.What is the smallest number you must add to 62 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  14. 14.What is the smallest number you must add to 230 to make it divisible by 8? (a) 1 (b) 6 (c) 2 (d) 3
  15. 15.What is the remainder when 86 is divided by 3? (a) 1 (b) 0 (c) 2
  16. 16.What is the smallest number you must add to 317 to make it divisible by 11? (a) 1 (b) 9 (c) 2 (d) 3
  17. 17.What is the units digit of 4^47? (a) 3 (b) 6 (c) 5 (d) 4
  18. 18.What is the smallest number you must add to 100 to make it divisible by 3? (a) 1 (b) 2
  19. 19.Which of these numbers is divisible by 6? (a) 2721 (b) 2725 (c) 2724 (d) 2732
  20. 20.Which of these numbers is divisible by 9? (a) 1515 (b) 1510 (c) 1512 (d) 1528
  21. 21.Which of these numbers is divisible by 11? (a) 9526 (b) 9529 (c) 9525 (d) 9547
  22. 22.Which of these numbers is divisible by 11? (a) 4055 (b) 4059 (c) 4072 (d) 4068
  23. 23.Which of these numbers is divisible by 9? (a) 616 (b) 612 (c) 607 (d) 623
  24. 24.What is the smallest number you must add to 162 to make it divisible by 8? (a) 2 (b) 7 (c) 5 (d) 6
  25. 25.What is the remainder when 640 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
  26. 26.What is the remainder when 122 is divided by 3? (a) 1 (b) 0 (c) 2
  27. 27.What is the smallest number you must add to 121 to make it divisible by 10? (a) 9 (b) 1 (c) 8 (d) 7
  28. 28.What is the smallest number you must add to 121 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 1
  29. 29.Which of these numbers is divisible by 8? (a) 5211 (b) 5226 (c) 5217 (d) 5216
  30. 30.Which of these numbers is divisible by 9? (a) 7190 (b) 7199 (c) 7182 (d) 7175
  31. 31.What is the units digit of 8^37? (a) 8 (b) 4 (c) 2 (d) 6
  32. 32.Which of these numbers is divisible by 9? (a) 1017 (b) 1014 (c) 1030 (d) 1024
  33. 33.Which of these numbers is divisible by 8? (a) 2368 (b) 2382 (c) 2370 (d) 2361
  34. 34.What is the units digit of 3^40? (a) 3 (b) 7 (c) 1 (d) 9
  35. 35.Which of these numbers is divisible by 3? (a) 2191 (b) 2188 (c) 2190 (d) 2194
  36. 36.What is the remainder when 348 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
  37. 37.What is the smallest number you must add to 226 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  38. 38.What is the units digit of 8^60? (a) 4 (b) 2 (c) 6 (d) 8
  39. 39.What is the smallest number you must add to 161 to make it divisible by 6? (a) 5 (b) 1 (c) 2 (d) 3
  40. 40.Which of these numbers is divisible by 4? (a) 3088 (b) 3089 (c) 3085 (d) 3093

Answer key

  1. 702
  2. 1
  3. 6985
  4. 1
  5. 3
  6. 1
  7. 6
  8. 4
  9. 1062
  10. 6
  11. 1
  12. 1
  13. 2
  14. 2
  15. 2
  16. 2
  17. 4
  18. 2
  19. 2724
  20. 1512
  21. 9526
  22. 4059
  23. 612
  24. 6
  25. 2
  26. 2
  27. 9
  28. 11
  29. 5216
  30. 7182
  31. 8
  32. 1017
  33. 2368
  34. 1
  35. 2190
  36. 8
  37. 4
  38. 6
  39. 1
  40. 3088

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