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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 9 · 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 311 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  2. 2.Which of these numbers is divisible by 6? (a) 4584 (b) 4585 (c) 4579 (d) 4592
  3. 3.What is the remainder when 173 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 6
  4. 4.What is the units digit of 2^10? (a) 4 (b) 6 (c) 2 (d) 8
  5. 5.What is the smallest number you must add to 49 to make it divisible by 3? (a) 1 (b) 2
  6. 6.Which of these numbers is divisible by 8? (a) 2174 (b) 2155 (c) 2160 (d) 2164
  7. 7.What is the smallest number you must add to 230 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  8. 8.What is the smallest number you must add to 41 to make it divisible by 3? (a) 1 (b) 2
  9. 9.What is the remainder when 168 is divided by 9? (a) 7 (b) 6 (c) 3 (d) 5
  10. 10.What is the smallest number you must add to 356 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
  11. 11.What is the smallest number you must add to 489 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
  12. 12.Which of these numbers is divisible by 9? (a) 3514 (b) 3510 (c) 3504 (d) 3527
  13. 13.What is the smallest number you must add to 158 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
  14. 14.What is the units digit of 9^14? (a) 0 (b) 9 (c) 1 (d) 2
  15. 15.What is the remainder when 304 is divided by 10? (a) 5 (b) 4 (c) 3 (d) 6
  16. 16.What is the smallest number you must add to 655 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
  17. 17.Which of these numbers is divisible by 4? (a) 1817 (b) 1823 (c) 1814 (d) 1816
  18. 18.What is the remainder when 172 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
  19. 19.What is the units digit of 2^11? (a) 8 (b) 2 (c) 6 (d) 4
  20. 20.Which of these numbers is divisible by 8? (a) 3928 (b) 3935 (c) 3940 (d) 3923
  21. 21.Which of these numbers is divisible by 11? (a) 2574 (b) 2594 (c) 2581 (d) 2564
  22. 22.Which of these numbers is divisible by 8? (a) 3446 (b) 3428 (c) 3433 (d) 3432
  23. 23.What is the remainder when 152 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
  24. 24.What is the remainder when 150 is divided by 8? (a) 5 (b) 7 (c) 6 (d) 2
  25. 25.What is the remainder when 404 is divided by 11? (a) 7 (b) 8 (c) 3 (d) 9
  26. 26.What is the remainder when 319 is divided by 7? (a) 3 (b) 5 (c) 6 (d) 4
  27. 27.What is the units digit of 2^36? (a) 4 (b) 8 (c) 6 (d) 2
  28. 28.What is the remainder when 277 is divided by 9? (a) 2 (b) 8 (c) 7 (d) 6
  29. 29.What is the remainder when 212 is divided by 10? (a) 2 (b) 8 (c) 3 (d) 1
  30. 30.What is the remainder when 509 is divided by 9? (a) 4 (b) 6 (c) 7 (d) 5
  31. 31.What is the remainder when 328 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
  32. 32.Which of these numbers is divisible by 8? (a) 3916 (b) 3929 (c) 3925 (d) 3920
  33. 33.What is the remainder when 889 is divided by 12? (a) 0 (b) 1 (c) 2 (d) 11
  34. 34.What is the smallest number you must add to 561 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 1
  35. 35.What is the smallest number you must add to 291 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  36. 36.What is the smallest number you must add to 549 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
  37. 37.Which of these numbers is divisible by 11? (a) 3041 (b) 3017 (c) 3025 (d) 3026
  38. 38.What is the smallest number you must add to 183 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
  39. 39.What is the smallest number you must add to 568 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
  40. 40.What is the smallest number you must add to 304 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4

Answer key

  1. 3
  2. 4584
  3. 5
  4. 4
  5. 2
  6. 2160
  7. 2
  8. 1
  9. 6
  10. 4
  11. 7
  12. 3510
  13. 4
  14. 1
  15. 4
  16. 5
  17. 1816
  18. 2
  19. 8
  20. 3928
  21. 2574
  22. 3432
  23. 2
  24. 6
  25. 8
  26. 4
  27. 6
  28. 7
  29. 2
  30. 5
  31. 3
  32. 3920
  33. 1
  34. 9
  35. 1
  36. 3
  37. 3025
  38. 3
  39. 8
  40. 1

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