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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 656 · 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 470 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
  2. 2.Which of these numbers is divisible by 4? (a) 857 (b) 852 (c) 850 (d) 854
  3. 3.What is the smallest number you must add to 555 to make it divisible by 12? (a) 10 (b) 8 (c) 3 (d) 9
  4. 4.What is the units digit of 9^56? (a) 1 (b) 0 (c) 2 (d) 9
  5. 5.What is the smallest number you must add to 358 to make it divisible by 11? (a) 6 (b) 5 (c) 4 (d) 7
  6. 6.What is the units digit of 8^35? (a) 2 (b) 8 (c) 6 (d) 4
  7. 7.Which of these numbers is divisible by 11? (a) 5774 (b) 5764 (c) 5757 (d) 5780
  8. 8.Which of these numbers is divisible by 11? (a) 2441 (b) 2426 (c) 2449 (d) 2431
  9. 9.What is the smallest number you must add to 270 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  10. 10.What is the remainder when 368 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
  11. 11.What is the units digit of 8^39? (a) 2 (b) 6 (c) 8 (d) 4
  12. 12.What is the smallest number you must add to 757 to make it divisible by 12? (a) 1 (b) 10 (c) 9 (d) 11
  13. 13.What is the smallest number you must add to 205 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 1
  14. 14.What is the smallest number you must add to 221 to make it divisible by 3? (a) 2 (b) 1
  15. 15.Which of these numbers is divisible by 8? (a) 5454 (b) 5471 (c) 5461 (d) 5456
  16. 16.What is the units digit of 8^36? (a) 4 (b) 8 (c) 6 (d) 2
  17. 17.What is the remainder when 147 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
  18. 18.What is the remainder when 237 is divided by 9? (a) 3 (b) 4 (c) 2 (d) 6
  19. 19.What is the smallest number you must add to 508 to make it divisible by 11? (a) 8 (b) 2 (c) 9 (d) 10
  20. 20.What is the smallest number you must add to 525 to make it divisible by 10? (a) 5 (b) 6 (c) 7 (d) 4
  21. 21.Which of these numbers is divisible by 3? (a) 2053 (b) 2059 (c) 2055 (d) 2056
  22. 22.What is the remainder when 221 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
  23. 23.What is the remainder when 97 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  24. 24.What is the units digit of 9^17? (a) 1 (b) 7 (c) 8 (d) 9
  25. 25.What is the remainder when 190 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2
  26. 26.What is the remainder when 795 is divided by 11? (a) 2 (b) 4 (c) 3 (d) 8
  27. 27.Which of these numbers is divisible by 11? (a) 3487 (b) 3499 (c) 3489 (d) 3481
  28. 28.What is the units digit of 7^12? (a) 1 (b) 3 (c) 9 (d) 7
  29. 29.What is the units digit of 7^46? (a) 7 (b) 1 (c) 3 (d) 9
  30. 30.What is the smallest number you must add to 367 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
  31. 31.What is the units digit of 9^28? (a) 2 (b) 9 (c) 1 (d) 0
  32. 32.What is the smallest number you must add to 452 to make it divisible by 7? (a) 3 (b) 5 (c) 2 (d) 4
  33. 33.What is the smallest number you must add to 98 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  34. 34.Which of these numbers is divisible by 4? (a) 3388 (b) 3390 (c) 3385 (d) 3393
  35. 35.What is the smallest number you must add to 349 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  36. 36.What is the smallest number you must add to 53 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  37. 37.What is the smallest number you must add to 148 to make it divisible by 10? (a) 3 (b) 2 (c) 8 (d) 1
  38. 38.What is the remainder when 794 is divided by 11? (a) 3 (b) 9 (c) 2 (d) 1
  39. 39.Which of these numbers is divisible by 11? (a) 8461 (b) 8448 (c) 8449 (d) 8439
  40. 40.What is the remainder when 442 is divided by 9? (a) 0 (b) 8 (c) 1 (d) 2

Answer key

  1. 1
  2. 852
  3. 9
  4. 1
  5. 5
  6. 2
  7. 5764
  8. 2431
  9. 2
  10. 5
  11. 2
  12. 11
  13. 11
  14. 1
  15. 5456
  16. 6
  17. 3
  18. 3
  19. 9
  20. 5
  21. 2055
  22. 1
  23. 1
  24. 9
  25. 1
  26. 3
  27. 3487
  28. 1
  29. 9
  30. 5
  31. 1
  32. 3
  33. 2
  34. 3388
  35. 1
  36. 3
  37. 2
  38. 2
  39. 8448
  40. 1

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