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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 398 · 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 smallest number you must add to 460 to make it divisible by 12? (a) 8 (b) 4 (c) 7 (d) 9
  2. 2.What is the smallest number you must add to 160 to make it divisible by 3? (a) 1 (b) 2
  3. 3.What is the remainder when 303 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
  4. 4.What is the remainder when 92 is divided by 3? (a) 2 (b) 1 (c) 0
  5. 5.Which of these numbers is divisible by 6? (a) 310 (b) 304 (c) 299 (d) 300
  6. 6.What is the units digit of 7^59? (a) 3 (b) 1 (c) 7 (d) 9
  7. 7.Which of these numbers is divisible by 3? (a) 629 (b) 631 (c) 630 (d) 634
  8. 8.What is the remainder when 203 is divided by 10? (a) 7 (b) 3 (c) 4 (d) 2
  9. 9.What is the units digit of 8^43? (a) 6 (b) 2 (c) 4 (d) 8
  10. 10.What is the remainder when 233 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
  11. 11.What is the remainder when 100 is divided by 9? (a) 1 (b) 0 (c) 2 (d) 8
  12. 12.What is the smallest number you must add to 601 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
  13. 13.What is the smallest number you must add to 773 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
  14. 14.What is the remainder when 485 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
  15. 15.What is the smallest number you must add to 465 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 3
  16. 16.What is the units digit of 7^53? (a) 1 (b) 7 (c) 3 (d) 9
  17. 17.What is the units digit of 2^19? (a) 2 (b) 6 (c) 8 (d) 4
  18. 18.What is the remainder when 304 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
  19. 19.Which of these numbers is divisible by 6? (a) 3178 (b) 3190 (c) 3183 (d) 3180
  20. 20.What is the smallest number you must add to 146 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
  21. 21.What is the units digit of 8^14? (a) 6 (b) 8 (c) 2 (d) 4
  22. 22.Which of these numbers is divisible by 11? (a) 8139 (b) 8129 (c) 8147 (d) 8120
  23. 23.Which of these numbers is divisible by 9? (a) 2658 (b) 2650 (c) 2639 (d) 2646
  24. 24.What is the units digit of 9^32? (a) 1 (b) 2 (c) 9 (d) 0
  25. 25.What is the smallest number you must add to 458 to make it divisible by 9? (a) 3 (b) 1 (c) 8 (d) 2
  26. 26.Which of these numbers is divisible by 11? (a) 5986 (b) 5979 (c) 5969 (d) 5973
  27. 27.Which of these numbers is divisible by 4? (a) 1447 (b) 1441 (c) 1439 (d) 1440
  28. 28.What is the smallest number you must add to 268 to make it divisible by 9? (a) 7 (b) 1 (c) 2 (d) 3
  29. 29.What is the units digit of 8^51? (a) 8 (b) 6 (c) 4 (d) 2
  30. 30.What is the smallest number you must add to 147 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  31. 31.What is the units digit of 7^32? (a) 9 (b) 1 (c) 7 (d) 3
  32. 32.Which of these numbers is divisible by 6? (a) 4412 (b) 4421 (c) 4425 (d) 4416
  33. 33.What is the remainder when 56 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
  34. 34.What is the smallest number you must add to 161 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  35. 35.What is the smallest number you must add to 293 to make it divisible by 8? (a) 2 (b) 4 (c) 3 (d) 5
  36. 36.Which of these numbers is divisible by 8? (a) 6477 (b) 6467 (c) 6482 (d) 6472
  37. 37.What is the units digit of 3^28? (a) 9 (b) 3 (c) 7 (d) 1
  38. 38.What is the smallest number you must add to 801 to make it divisible by 12? (a) 2 (b) 9 (c) 3 (d) 4
  39. 39.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
  40. 40.What is the smallest number you must add to 280 to make it divisible by 12? (a) 8 (b) 9 (c) 4 (d) 7

Answer key

  1. 8
  2. 2
  3. 3
  4. 2
  5. 300
  6. 3
  7. 630
  8. 3
  9. 2
  10. 5
  11. 1
  12. 2
  13. 7
  14. 5
  15. 4
  16. 7
  17. 8
  18. 4
  19. 3180
  20. 4
  21. 4
  22. 8129
  23. 2646
  24. 1
  25. 1
  26. 5973
  27. 1440
  28. 2
  29. 2
  30. 3
  31. 1
  32. 4416
  33. 1
  34. 4
  35. 3
  36. 6472
  37. 1
  38. 3
  39. 1
  40. 8

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