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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 153 · 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 51 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  2. 2.Which of these numbers is divisible by 3? (a) 560 (b) 558 (c) 556 (d) 563
  3. 3.Which of these numbers is divisible by 3? (a) 1466 (b) 1472 (c) 1469 (d) 1467
  4. 4.What is the remainder when 304 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 2
  5. 5.Which of these numbers is divisible by 11? (a) 1363 (b) 1349 (c) 1353 (d) 1367
  6. 6.What is the remainder when 418 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
  7. 7.Which of these numbers is divisible by 8? (a) 561 (b) 548 (c) 558 (d) 552
  8. 8.Which of these numbers is divisible by 3? (a) 2614 (b) 2611 (c) 2613 (d) 2617
  9. 9.What is the smallest number you must add to 59 to make it divisible by 3? (a) 2 (b) 1
  10. 10.Which of these numbers is divisible by 4? (a) 2602 (b) 2606 (c) 2611 (d) 2604
  11. 11.What is the remainder when 64 is divided by 3? (a) 2 (b) 1 (c) 0
  12. 12.What is the units digit of 4^46? (a) 7 (b) 6 (c) 5 (d) 4
  13. 13.What is the smallest number you must add to 121 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 4
  14. 14.What is the smallest number you must add to 70 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  15. 15.What is the smallest number you must add to 383 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 5
  16. 16.What is the smallest number you must add to 443 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
  17. 17.Which of these numbers is divisible by 11? (a) 3680 (b) 3672 (c) 3663 (d) 3657
  18. 18.Which of these numbers is divisible by 9? (a) 4454 (b) 4460 (c) 4446 (d) 4441
  19. 19.What is the remainder when 452 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
  20. 20.What is the smallest number you must add to 205 to make it divisible by 3? (a) 1 (b) 2
  21. 21.What is the remainder when 767 is divided by 11? (a) 7 (b) 8 (c) 9 (d) 3
  22. 22.What is the smallest number you must add to 296 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
  23. 23.What is the smallest number you must add to 477 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 1
  24. 24.What is the units digit of 4^39? (a) 5 (b) 6 (c) 4 (d) 3
  25. 25.What is the smallest number you must add to 715 to make it divisible by 12? (a) 4 (b) 6 (c) 7 (d) 5
  26. 26.What is the smallest number you must add to 314 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  27. 27.What is the smallest number you must add to 613 to make it divisible by 8? (a) 4 (b) 3 (c) 2 (d) 5
  28. 28.What is the units digit of 7^50? (a) 3 (b) 7 (c) 9 (d) 1
  29. 29.Which of these numbers is divisible by 9? (a) 3728 (b) 3724 (c) 3717 (d) 3711
  30. 30.What is the remainder when 401 is divided by 10? (a) 9 (b) 0 (c) 2 (d) 1
  31. 31.What is the units digit of 8^42? (a) 8 (b) 2 (c) 4 (d) 6
  32. 32.What is the smallest number you must add to 523 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
  33. 33.Which of these numbers is divisible by 11? (a) 9603 (b) 9602 (c) 9610 (d) 9616
  34. 34.What is the smallest number you must add to 172 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 1
  35. 35.Which of these numbers is divisible by 8? (a) 1108 (b) 1119 (c) 1102 (d) 1104
  36. 36.What is the remainder when 566 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
  37. 37.Which of these numbers is divisible by 4? (a) 838 (b) 832 (c) 833 (d) 830
  38. 38.What is the remainder when 279 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
  39. 39.Which of these numbers is divisible by 3? (a) 2077 (b) 2075 (c) 2081 (d) 2076
  40. 40.What is the units digit of 8^31? (a) 6 (b) 2 (c) 4 (d) 8

Answer key

  1. 1
  2. 558
  3. 1467
  4. 3
  5. 1353
  6. 4
  7. 552
  8. 2613
  9. 1
  10. 2604
  11. 1
  12. 6
  13. 5
  14. 2
  15. 2
  16. 1
  17. 3663
  18. 4446
  19. 2
  20. 2
  21. 8
  22. 4
  23. 6
  24. 4
  25. 5
  26. 2
  27. 3
  28. 9
  29. 3717
  30. 1
  31. 4
  32. 5
  33. 9603
  34. 2
  35. 1104
  36. 6
  37. 832
  38. 4
  39. 2076
  40. 2

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