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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 391 · 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 450 to make it divisible by 11? (a) 1 (b) 10 (c) 2 (d) 3
  2. 2.Which of these numbers is divisible by 8? (a) 2912 (b) 2908 (c) 2917 (d) 2927
  3. 3.What is the smallest number you must add to 494 to make it divisible by 8? (a) 6 (b) 1 (c) 2 (d) 3
  4. 4.What is the smallest number you must add to 441 to make it divisible by 12? (a) 4 (b) 9 (c) 3 (d) 2
  5. 5.What is the units digit of 9^20? (a) 9 (b) 0 (c) 1 (d) 2
  6. 6.What is the smallest number you must add to 169 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  7. 7.Which of these numbers is divisible by 8? (a) 5120 (b) 5115 (c) 5133 (d) 5126
  8. 8.What is the remainder when 371 is divided by 12? (a) 10 (b) 11 (c) 1 (d) 9
  9. 9.What is the units digit of 3^30? (a) 1 (b) 7 (c) 3 (d) 9
  10. 10.Which of these numbers is divisible by 4? (a) 2430 (b) 2427 (c) 2428 (d) 2435
  11. 11.What is the smallest number you must add to 241 to make it divisible by 12? (a) 10 (b) 11 (c) 9 (d) 1
  12. 12.What is the units digit of 8^43? (a) 4 (b) 8 (c) 6 (d) 2
  13. 13.What is the units digit of 8^38? (a) 6 (b) 2 (c) 4 (d) 8
  14. 14.What is the smallest number you must add to 383 to make it divisible by 6? (a) 1 (b) 2 (c) 5 (d) 3
  15. 15.Which of these numbers is divisible by 6? (a) 1032 (b) 1043 (c) 1037 (d) 1031
  16. 16.What is the remainder when 157 is divided by 3? (a) 0 (b) 2 (c) 1
  17. 17.Which of these numbers is divisible by 6? (a) 970 (b) 960 (c) 958 (d) 963
  18. 18.Which of these numbers is divisible by 6? (a) 1738 (b) 1745 (c) 1733 (d) 1734
  19. 19.What is the smallest number you must add to 98 to make it divisible by 3? (a) 1 (b) 2
  20. 20.Which of these numbers is divisible by 6? (a) 4586 (b) 4581 (c) 4595 (d) 4584
  21. 21.Which of these numbers is divisible by 8? (a) 2840 (b) 2855 (c) 2841 (d) 2835
  22. 22.What is the remainder when 263 is divided by 8? (a) 1 (b) 6 (c) 7 (d) 5
  23. 23.Which of these numbers is divisible by 11? (a) 4716 (b) 4698 (c) 4729 (d) 4708
  24. 24.What is the remainder when 766 is divided by 11? (a) 6 (b) 8 (c) 7 (d) 4
  25. 25.What is the smallest number you must add to 146 to make it divisible by 11? (a) 8 (b) 3 (c) 9 (d) 7
  26. 26.What is the remainder when 79 is divided by 3? (a) 0 (b) 2 (c) 1
  27. 27.Which of these numbers is divisible by 8? (a) 1513 (b) 1525 (c) 1520 (d) 1534
  28. 28.What is the remainder when 233 is divided by 7? (a) 3 (b) 2 (c) 1 (d) 5
  29. 29.What is the smallest number you must add to 303 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  30. 30.What is the units digit of 9^16? (a) 9 (b) 1 (c) 2 (d) 0
  31. 31.Which of these numbers is divisible by 11? (a) 9791 (b) 9801 (c) 9820 (d) 9809
  32. 32.What is the units digit of 2^36? (a) 2 (b) 8 (c) 6 (d) 4
  33. 33.What is the units digit of 8^40? (a) 4 (b) 6 (c) 2 (d) 8
  34. 34.What is the smallest number you must add to 305 to make it divisible by 12? (a) 7 (b) 8 (c) 6 (d) 5
  35. 35.What is the smallest number you must add to 922 to make it divisible by 12? (a) 2 (b) 1 (c) 10 (d) 3
  36. 36.What is the smallest number you must add to 167 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  37. 37.What is the smallest number you must add to 170 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
  38. 38.What is the units digit of 4^10? (a) 4 (b) 5 (c) 7 (d) 6
  39. 39.Which of these numbers is divisible by 3? (a) 2015 (b) 2009 (c) 2012 (d) 2010
  40. 40.What is the smallest number you must add to 114 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3

Answer key

  1. 1
  2. 2912
  3. 2
  4. 3
  5. 1
  6. 1
  7. 5120
  8. 11
  9. 9
  10. 2428
  11. 11
  12. 2
  13. 4
  14. 1
  15. 1032
  16. 1
  17. 960
  18. 1734
  19. 1
  20. 4584
  21. 2840
  22. 7
  23. 4708
  24. 7
  25. 8
  26. 1
  27. 1520
  28. 2
  29. 1
  30. 1
  31. 9801
  32. 6
  33. 6
  34. 7
  35. 2
  36. 1
  37. 4
  38. 6
  39. 2010
  40. 1

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