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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 963 · 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 178 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  2. 2.What is the smallest number you must add to 56 to make it divisible by 3? (a) 1 (b) 2
  3. 3.Which of these numbers is divisible by 8? (a) 6362 (b) 6368 (c) 6369 (d) 6383
  4. 4.What is the remainder when 89 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
  5. 5.What is the remainder when 547 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
  6. 6.What is the remainder when 213 is divided by 12? (a) 9 (b) 3 (c) 8 (d) 10
  7. 7.What is the smallest number you must add to 73 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  8. 8.Which of these numbers is divisible by 8? (a) 471 (b) 472 (c) 479 (d) 481
  9. 9.What is the smallest number you must add to 715 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
  10. 10.What is the smallest number you must add to 361 to make it divisible by 12? (a) 10 (b) 9 (c) 1 (d) 11
  11. 11.Which of these numbers is divisible by 3? (a) 931 (b) 925 (c) 927 (d) 929
  12. 12.What is the units digit of 3^46? (a) 9 (b) 1 (c) 3 (d) 7
  13. 13.What is the smallest number you must add to 854 to make it divisible by 11? (a) 5 (b) 4 (c) 3 (d) 7
  14. 14.What is the smallest number you must add to 44 to make it divisible by 3? (a) 1 (b) 2
  15. 15.Which of these numbers is divisible by 11? (a) 4811 (b) 4807 (c) 4801 (d) 4824
  16. 16.Which of these numbers is divisible by 4? (a) 2740 (b) 2741 (c) 2737 (d) 2745
  17. 17.What is the remainder when 74 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  18. 18.Which of these numbers is divisible by 3? (a) 2651 (b) 2648 (c) 2653 (d) 2649
  19. 19.What is the units digit of 9^33? (a) 9 (b) 7 (c) 1 (d) 8
  20. 20.What is the units digit of 9^43? (a) 9 (b) 1 (c) 7 (d) 8
  21. 21.What is the smallest number you must add to 99 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  22. 22.Which of these numbers is divisible by 4? (a) 1141 (b) 1136 (c) 1133 (d) 1138
  23. 23.What is the remainder when 446 is divided by 12? (a) 10 (b) 2 (c) 3 (d) 1
  24. 24.What is the units digit of 7^34? (a) 3 (b) 9 (c) 1 (d) 7
  25. 25.Which of these numbers is divisible by 8? (a) 5584 (b) 5594 (c) 5590 (d) 5577
  26. 26.What is the smallest number you must add to 401 to make it divisible by 7? (a) 6 (b) 2 (c) 5 (d) 4
  27. 27.What is the units digit of 4^24? (a) 5 (b) 7 (c) 6 (d) 4
  28. 28.Which of these numbers is divisible by 8? (a) 3322 (b) 3318 (c) 3320 (d) 3329
  29. 29.What is the remainder when 200 is divided by 3? (a) 1 (b) 0 (c) 2
  30. 30.Which of these numbers is divisible by 9? (a) 2239 (b) 2245 (c) 2251 (d) 2241
  31. 31.What is the smallest number you must add to 274 to make it divisible by 11? (a) 3 (b) 2 (c) 10 (d) 1
  32. 32.What is the smallest number you must add to 163 to make it divisible by 7? (a) 5 (b) 6 (c) 2 (d) 4
  33. 33.What is the remainder when 67 is divided by 3? (a) 1 (b) 0 (c) 2
  34. 34.What is the smallest number you must add to 108 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 3
  35. 35.Which of these numbers is divisible by 8? (a) 1889 (b) 1896 (c) 1897 (d) 1911
  36. 36.Which of these numbers is divisible by 11? (a) 3462 (b) 3436 (c) 3443 (d) 3446
  37. 37.What is the remainder when 223 is divided by 12? (a) 8 (b) 7 (c) 5 (d) 6
  38. 38.Which of these numbers is divisible by 4? (a) 2677 (b) 2670 (c) 2672 (d) 2673
  39. 39.Which of these numbers is divisible by 11? (a) 8064 (b) 8063 (c) 8056 (d) 8076
  40. 40.What is the smallest number you must add to 213 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4

Answer key

  1. 2
  2. 1
  3. 6368
  4. 1
  5. 1
  6. 9
  7. 3
  8. 472
  9. 5
  10. 11
  11. 927
  12. 9
  13. 4
  14. 1
  15. 4807
  16. 2740
  17. 4
  18. 2649
  19. 9
  20. 9
  21. 1
  22. 1136
  23. 2
  24. 9
  25. 5584
  26. 5
  27. 6
  28. 3320
  29. 2
  30. 2241
  31. 1
  32. 5
  33. 1
  34. 4
  35. 1896
  36. 3443
  37. 7
  38. 2672
  39. 8063
  40. 3

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