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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 668 · 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 9? (a) 1212 (b) 1228 (c) 1215 (d) 1221
  2. 2.Which of these numbers is divisible by 9? (a) 5354 (b) 5332 (c) 5339 (d) 5337
  3. 3.What is the smallest number you must add to 701 to make it divisible by 10? (a) 9 (b) 8 (c) 1 (d) 7
  4. 4.What is the smallest number you must add to 215 to make it divisible by 6? (a) 2 (b) 5 (c) 1 (d) 3
  5. 5.Which of these numbers is divisible by 6? (a) 4090 (b) 4092 (c) 4101 (d) 4097
  6. 6.What is the units digit of 4^21? (a) 5 (b) 3 (c) 4 (d) 6
  7. 7.What is the remainder when 61 is divided by 3? (a) 0 (b) 1 (c) 2
  8. 8.What is the smallest number you must add to 361 to make it divisible by 11? (a) 9 (b) 3 (c) 1 (d) 2
  9. 9.Which of these numbers is divisible by 6? (a) 4850 (b) 4856 (c) 4848 (d) 4847
  10. 10.What is the remainder when 594 is divided by 10? (a) 3 (b) 6 (c) 5 (d) 4
  11. 11.What is the remainder when 334 is divided by 11? (a) 7 (b) 3 (c) 5 (d) 4
  12. 12.Which of these numbers is divisible by 4? (a) 2552 (b) 2551 (c) 2555 (d) 2557
  13. 13.What is the smallest number you must add to 466 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
  14. 14.Which of these numbers is divisible by 6? (a) 2413 (b) 2418 (c) 2425 (d) 2420
  15. 15.What is the smallest number you must add to 897 to make it divisible by 12? (a) 4 (b) 9 (c) 2 (d) 3
  16. 16.What is the smallest number you must add to 317 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  17. 17.Which of these numbers is divisible by 8? (a) 3881 (b) 3872 (c) 3877 (d) 3871
  18. 18.Which of these numbers is divisible by 9? (a) 5517 (b) 5510 (c) 5530 (d) 5521
  19. 19.What is the smallest number you must add to 113 to make it divisible by 8? (a) 1 (b) 7 (c) 5 (d) 6
  20. 20.What is the remainder when 691 is divided by 9? (a) 2 (b) 7 (c) 8 (d) 6
  21. 21.What is the smallest number you must add to 573 to make it divisible by 11? (a) 8 (b) 10 (c) 9 (d) 1
  22. 22.What is the units digit of 3^41? (a) 1 (b) 3 (c) 7 (d) 9
  23. 23.What is the units digit of 8^37? (a) 4 (b) 6 (c) 8 (d) 2
  24. 24.Which of these numbers is divisible by 6? (a) 1217 (b) 1223 (c) 1207 (d) 1212
  25. 25.What is the remainder when 333 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
  26. 26.Which of these numbers is divisible by 11? (a) 1997 (b) 1989 (c) 1980 (d) 1979
  27. 27.What is the units digit of 8^41? (a) 6 (b) 8 (c) 4 (d) 2
  28. 28.What is the units digit of 7^11? (a) 1 (b) 3 (c) 9 (d) 7
  29. 29.Which of these numbers is divisible by 4? (a) 2804 (b) 2807 (c) 2811 (d) 2803
  30. 30.What is the units digit of 4^17? (a) 6 (b) 5 (c) 4 (d) 3
  31. 31.What is the remainder when 271 is divided by 12? (a) 7 (b) 8 (c) 5 (d) 6
  32. 32.Which of these numbers is divisible by 11? (a) 8273 (b) 8251 (c) 8269 (d) 8261
  33. 33.What is the remainder when 459 is divided by 12? (a) 9 (b) 2 (c) 4 (d) 3
  34. 34.What is the smallest number you must add to 339 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
  35. 35.Which of these numbers is divisible by 3? (a) 309 (b) 310 (c) 314 (d) 308
  36. 36.What is the units digit of 8^45? (a) 4 (b) 8 (c) 2 (d) 6
  37. 37.What is the smallest number you must add to 498 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 1
  38. 38.What is the smallest number you must add to 311 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 1
  39. 39.What is the units digit of 9^33? (a) 9 (b) 8 (c) 1 (d) 7
  40. 40.What is the smallest number you must add to 273 to make it divisible by 12? (a) 9 (b) 3 (c) 2 (d) 4

Answer key

  1. 1215
  2. 5337
  3. 9
  4. 1
  5. 4092
  6. 4
  7. 1
  8. 2
  9. 4848
  10. 4
  11. 4
  12. 2552
  13. 2
  14. 2418
  15. 3
  16. 3
  17. 3872
  18. 5517
  19. 7
  20. 7
  21. 10
  22. 3
  23. 8
  24. 1212
  25. 3
  26. 1980
  27. 8
  28. 3
  29. 2804
  30. 4
  31. 7
  32. 8261
  33. 3
  34. 4
  35. 309
  36. 8
  37. 6
  38. 1
  39. 9
  40. 3

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