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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 247 · 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.Which of these numbers is divisible by 11? (a) 7206 (b) 7219 (c) 7216 (d) 7235
  2. 2.What is the remainder when 125 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
  3. 3.What is the remainder when 166 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
  4. 4.Which of these numbers is divisible by 4? (a) 730 (b) 723 (c) 726 (d) 724
  5. 5.Which of these numbers is divisible by 4? (a) 358 (b) 363 (c) 353 (d) 356
  6. 6.Which of these numbers is divisible by 9? (a) 3153 (b) 3145 (c) 3150 (d) 3165
  7. 7.What is the units digit of 2^21? (a) 2 (b) 6 (c) 4 (d) 8
  8. 8.Which of these numbers is divisible by 9? (a) 1117 (b) 1106 (c) 1107 (d) 1115
  9. 9.What is the units digit of 2^31? (a) 6 (b) 8 (c) 2 (d) 4
  10. 10.What is the smallest number you must add to 242 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 2
  11. 11.What is the smallest number you must add to 227 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  12. 12.What is the remainder when 582 is divided by 11? (a) 10 (b) 9 (c) 8 (d) 1
  13. 13.Which of these numbers is divisible by 6? (a) 4235 (b) 4243 (c) 4241 (d) 4236
  14. 14.Which of these numbers is divisible by 6? (a) 1431 (b) 1434 (c) 1444 (d) 1437
  15. 15.What is the smallest number you must add to 673 to make it divisible by 10? (a) 8 (b) 3 (c) 7 (d) 6
  16. 16.Which of these numbers is divisible by 8? (a) 584 (b) 578 (c) 591 (d) 594
  17. 17.What is the remainder when 729 is divided by 10? (a) 8 (b) 7 (c) 1 (d) 9
  18. 18.What is the smallest number you must add to 233 to make it divisible by 3? (a) 2 (b) 1
  19. 19.What is the units digit of 8^20? (a) 4 (b) 2 (c) 8 (d) 6
  20. 20.What is the remainder when 206 is divided by 11? (a) 3 (b) 9 (c) 8 (d) 7
  21. 21.Which of these numbers is divisible by 9? (a) 942 (b) 921 (c) 927 (d) 929
  22. 22.Which of these numbers is divisible by 3? (a) 1797 (b) 1801 (c) 1795 (d) 1799
  23. 23.What is the smallest number you must add to 744 to make it divisible by 10? (a) 5 (b) 6 (c) 7 (d) 4
  24. 24.What is the remainder when 182 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  25. 25.What is the smallest number you must add to 848 to make it divisible by 11? (a) 9 (b) 1 (c) 10 (d) 8
  26. 26.What is the remainder when 501 is divided by 11? (a) 7 (b) 5 (c) 8 (d) 6
  27. 27.Which of these numbers is divisible by 4? (a) 422 (b) 418 (c) 427 (d) 420
  28. 28.What is the units digit of 7^36? (a) 1 (b) 9 (c) 3 (d) 7
  29. 29.Which of these numbers is divisible by 11? (a) 1427 (b) 1413 (c) 1433 (d) 1419
  30. 30.What is the remainder when 85 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
  31. 31.What is the units digit of 3^40? (a) 1 (b) 9 (c) 7 (d) 3
  32. 32.Which of these numbers is divisible by 3? (a) 418 (b) 422 (c) 415 (d) 417
  33. 33.What is the smallest number you must add to 803 to make it divisible by 12? (a) 3 (b) 11 (c) 1 (d) 2
  34. 34.What is the smallest number you must add to 278 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  35. 35.What is the smallest number you must add to 134 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
  36. 36.What is the smallest number you must add to 501 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 2
  37. 37.What is the remainder when 485 is divided by 7? (a) 3 (b) 5 (c) 1 (d) 2
  38. 38.What is the units digit of 3^15? (a) 1 (b) 9 (c) 3 (d) 7
  39. 39.What is the smallest number you must add to 308 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
  40. 40.What is the smallest number you must add to 87 to make it divisible by 4? (a) 2 (b) 3 (c) 1

Answer key

  1. 7216
  2. 5
  3. 1
  4. 724
  5. 356
  6. 3150
  7. 2
  8. 1107
  9. 8
  10. 3
  11. 1
  12. 10
  13. 4236
  14. 1434
  15. 7
  16. 584
  17. 9
  18. 1
  19. 6
  20. 8
  21. 927
  22. 1797
  23. 6
  24. 2
  25. 10
  26. 6
  27. 420
  28. 1
  29. 1419
  30. 1
  31. 1
  32. 417
  33. 1
  34. 2
  35. 1
  36. 3
  37. 2
  38. 7
  39. 4
  40. 1

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