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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 638 · 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 436 is divided by 10? (a) 5 (b) 4 (c) 6 (d) 7
  2. 2.Which of these numbers is divisible by 11? (a) 7129 (b) 7141 (c) 7128 (d) 7126
  3. 3.What is the units digit of 7^40? (a) 1 (b) 9 (c) 7 (d) 3
  4. 4.What is the units digit of 3^59? (a) 3 (b) 9 (c) 7 (d) 1
  5. 5.What is the remainder when 409 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 3
  6. 6.What is the remainder when 227 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
  7. 7.What is the remainder when 436 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
  8. 8.Which of these numbers is divisible by 4? (a) 2281 (b) 2280 (c) 2279 (d) 2286
  9. 9.What is the units digit of 2^26? (a) 2 (b) 8 (c) 4 (d) 6
  10. 10.Which of these numbers is divisible by 6? (a) 4710 (b) 4720 (c) 4706 (d) 4713
  11. 11.Which of these numbers is divisible by 6? (a) 258 (b) 260 (c) 269 (d) 255
  12. 12.What is the smallest number you must add to 53 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  13. 13.What is the smallest number you must add to 125 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  14. 14.What is the smallest number you must add to 697 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 4
  15. 15.What is the smallest number you must add to 261 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  16. 16.Which of these numbers is divisible by 11? (a) 7451 (b) 7445 (c) 7459 (d) 7447
  17. 17.What is the smallest number you must add to 268 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 2
  18. 18.What is the units digit of 2^14? (a) 2 (b) 6 (c) 8 (d) 4
  19. 19.What is the smallest number you must add to 399 to make it divisible by 8? (a) 1 (b) 2 (c) 3 (d) 7
  20. 20.Which of these numbers is divisible by 8? (a) 4805 (b) 4823 (c) 4808 (d) 4811
  21. 21.What is the remainder when 91 is divided by 8? (a) 4 (b) 2 (c) 3 (d) 5
  22. 22.What is the smallest number you must add to 281 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  23. 23.Which of these numbers is divisible by 11? (a) 8701 (b) 8693 (c) 8705 (d) 8713
  24. 24.What is the remainder when 174 is divided by 8? (a) 6 (b) 7 (c) 2 (d) 5
  25. 25.Which of these numbers is divisible by 3? (a) 2690 (b) 2693 (c) 2695 (d) 2691
  26. 26.What is the remainder when 365 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
  27. 27.Which of these numbers is divisible by 4? (a) 1501 (b) 1496 (c) 1495 (d) 1497
  28. 28.What is the smallest number you must add to 278 to make it divisible by 9? (a) 8 (b) 2 (c) 3 (d) 1
  29. 29.What is the smallest number you must add to 92 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
  30. 30.What is the smallest number you must add to 321 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  31. 31.What is the remainder when 245 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
  32. 32.Which of these numbers is divisible by 6? (a) 1887 (b) 1884 (c) 1893 (d) 1883
  33. 33.What is the units digit of 2^18? (a) 8 (b) 4 (c) 2 (d) 6
  34. 34.Which of these numbers is divisible by 6? (a) 4431 (b) 4422 (c) 4418 (d) 4427
  35. 35.What is the smallest number you must add to 789 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 9
  36. 36.Which of these numbers is divisible by 3? (a) 1625 (b) 1627 (c) 1623 (d) 1622
  37. 37.What is the remainder when 537 is divided by 11? (a) 10 (b) 2 (c) 8 (d) 9
  38. 38.What is the units digit of 2^19? (a) 4 (b) 6 (c) 2 (d) 8
  39. 39.What is the remainder when 118 is divided by 3? (a) 0 (b) 2 (c) 1
  40. 40.What is the units digit of 2^30? (a) 2 (b) 4 (c) 6 (d) 8

Answer key

  1. 6
  2. 7128
  3. 1
  4. 7
  5. 4
  6. 2
  7. 4
  8. 2280
  9. 4
  10. 4710
  11. 258
  12. 2
  13. 3
  14. 5
  15. 3
  16. 7447
  17. 5
  18. 4
  19. 1
  20. 4808
  21. 3
  22. 4
  23. 8701
  24. 6
  25. 2691
  26. 5
  27. 1496
  28. 1
  29. 6
  30. 3
  31. 1
  32. 1884
  33. 4
  34. 4422
  35. 1
  36. 1623
  37. 9
  38. 8
  39. 1
  40. 4

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