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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 470 · 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 7^53? (a) 9 (b) 1 (c) 3 (d) 7
  2. 2.Which of these numbers is divisible by 8? (a) 1267 (b) 1274 (c) 1264 (d) 1259
  3. 3.Which of these numbers is divisible by 8? (a) 5957 (b) 5944 (c) 5937 (d) 5947
  4. 4.Which of these numbers is divisible by 3? (a) 1533 (b) 1532 (c) 1537 (d) 1534
  5. 5.What is the smallest number you must add to 390 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 5
  6. 6.What is the remainder when 282 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  7. 7.What is the remainder when 118 is divided by 3? (a) 0 (b) 1 (c) 2
  8. 8.Which of these numbers is divisible by 4? (a) 787 (b) 783 (c) 784 (d) 790
  9. 9.What is the remainder when 317 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
  10. 10.Which of these numbers is divisible by 3? (a) 2278 (b) 2277 (c) 2276 (d) 2282
  11. 11.Which of these numbers is divisible by 11? (a) 6267 (b) 6270 (c) 6271 (d) 6290
  12. 12.What is the smallest number you must add to 489 to make it divisible by 9? (a) 7 (b) 3 (c) 5 (d) 6
  13. 13.Which of these numbers is divisible by 8? (a) 5078 (b) 5066 (c) 5086 (d) 5072
  14. 14.What is the remainder when 808 is divided by 10? (a) 9 (b) 2 (c) 7 (d) 8
  15. 15.What is the remainder when 708 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
  16. 16.What is the smallest number you must add to 117 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  17. 17.What is the smallest number you must add to 98 to make it divisible by 3? (a) 2 (b) 1
  18. 18.What is the units digit of 8^37? (a) 4 (b) 6 (c) 8 (d) 2
  19. 19.What is the smallest number you must add to 251 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
  20. 20.What is the smallest number you must add to 668 to make it divisible by 10? (a) 8 (b) 3 (c) 2 (d) 1
  21. 21.Which of these numbers is divisible by 4? (a) 2977 (b) 2975 (c) 2981 (d) 2976
  22. 22.Which of these numbers is divisible by 9? (a) 1431 (b) 1446 (c) 1429 (d) 1432
  23. 23.What is the units digit of 2^60? (a) 8 (b) 2 (c) 4 (d) 6
  24. 24.What is the remainder when 275 is divided by 9? (a) 4 (b) 7 (c) 6 (d) 5
  25. 25.What is the remainder when 401 is divided by 9? (a) 7 (b) 6 (c) 4 (d) 5
  26. 26.What is the smallest number you must add to 551 to make it divisible by 11? (a) 1 (b) 10 (c) 8 (d) 9
  27. 27.Which of these numbers is divisible by 9? (a) 4741 (b) 4734 (c) 4728 (d) 4749
  28. 28.What is the smallest number you must add to 212 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2
  29. 29.What is the units digit of 3^57? (a) 7 (b) 9 (c) 1 (d) 3
  30. 30.Which of these numbers is divisible by 9? (a) 1690 (b) 1692 (c) 1694 (d) 1705
  31. 31.Which of these numbers is divisible by 8? (a) 5705 (b) 5704 (c) 5717 (d) 5700
  32. 32.What is the units digit of 9^15? (a) 8 (b) 7 (c) 9 (d) 1
  33. 33.What is the units digit of 4^59? (a) 5 (b) 6 (c) 4 (d) 3
  34. 34.Which of these numbers is divisible by 4? (a) 1471 (b) 1466 (c) 1473 (d) 1468
  35. 35.Which of these numbers is divisible by 8? (a) 4105 (b) 4113 (c) 4099 (d) 4104
  36. 36.What is the remainder when 579 is divided by 12? (a) 2 (b) 3 (c) 4 (d) 9
  37. 37.What is the remainder when 320 is divided by 6? (a) 2 (b) 1 (c) 3 (d) 4
  38. 38.What is the smallest number you must add to 433 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 5
  39. 39.What is the smallest number you must add to 138 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  40. 40.What is the smallest number you must add to 393 to make it divisible by 10? (a) 8 (b) 6 (c) 3 (d) 7

Answer key

  1. 7
  2. 1264
  3. 5944
  4. 1533
  5. 2
  6. 2
  7. 1
  8. 784
  9. 1
  10. 2277
  11. 6270
  12. 6
  13. 5072
  14. 8
  15. 6
  16. 3
  17. 1
  18. 8
  19. 4
  20. 2
  21. 2976
  22. 1431
  23. 6
  24. 5
  25. 5
  26. 10
  27. 4734
  28. 5
  29. 3
  30. 1692
  31. 5704
  32. 9
  33. 4
  34. 1468
  35. 4104
  36. 3
  37. 2
  38. 5
  39. 2
  40. 7

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