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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 417 · 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 488 is divided by 9? (a) 2 (b) 1 (c) 7 (d) 3
  2. 2.What is the remainder when 133 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
  3. 3.What is the smallest number you must add to 427 to make it divisible by 10? (a) 4 (b) 3 (c) 2 (d) 7
  4. 4.What is the smallest number you must add to 401 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  5. 5.What is the smallest number you must add to 133 to make it divisible by 12? (a) 11 (b) 1 (c) 10 (d) 9
  6. 6.Which of these numbers is divisible by 3? (a) 471 (b) 476 (c) 470 (d) 472
  7. 7.Which of these numbers is divisible by 6? (a) 1115 (b) 1104 (c) 1101 (d) 1105
  8. 8.What is the smallest number you must add to 226 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  9. 9.What is the units digit of 8^36? (a) 4 (b) 6 (c) 2 (d) 8
  10. 10.What is the smallest number you must add to 189 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
  11. 11.What is the units digit of 3^57? (a) 3 (b) 1 (c) 7 (d) 9
  12. 12.What is the units digit of 3^43? (a) 1 (b) 7 (c) 9 (d) 3
  13. 13.What is the smallest number you must add to 119 to make it divisible by 8? (a) 3 (b) 2 (c) 7 (d) 1
  14. 14.What is the smallest number you must add to 418 to make it divisible by 8? (a) 7 (b) 2 (c) 5 (d) 6
  15. 15.What is the smallest number you must add to 63 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  16. 16.What is the smallest number you must add to 786 to make it divisible by 12? (a) 8 (b) 5 (c) 7 (d) 6
  17. 17.What is the smallest number you must add to 283 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 4
  18. 18.What is the smallest number you must add to 223 to make it divisible by 3? (a) 2 (b) 1
  19. 19.What is the units digit of 4^32? (a) 6 (b) 7 (c) 4 (d) 5
  20. 20.What is the smallest number you must add to 113 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  21. 21.Which of these numbers is divisible by 4? (a) 519 (b) 522 (c) 516 (d) 514
  22. 22.Which of these numbers is divisible by 9? (a) 592 (b) 584 (c) 585 (d) 601
  23. 23.What is the smallest number you must add to 154 to make it divisible by 3? (a) 1 (b) 2
  24. 24.What is the smallest number you must add to 284 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
  25. 25.Which of these numbers is divisible by 3? (a) 1423 (b) 1420 (c) 1419 (d) 1418
  26. 26.What is the remainder when 383 is divided by 8? (a) 1 (b) 7 (c) 5 (d) 6
  27. 27.What is the units digit of 7^59? (a) 7 (b) 1 (c) 3 (d) 9
  28. 28.What is the smallest number you must add to 283 to make it divisible by 11? (a) 8 (b) 4 (c) 2 (d) 3
  29. 29.Which of these numbers is divisible by 3? (a) 913 (b) 912 (c) 910 (d) 917
  30. 30.What is the units digit of 2^44? (a) 4 (b) 2 (c) 8 (d) 6
  31. 31.What is the remainder when 317 is divided by 6? (a) 5 (b) 4 (c) 1 (d) 3
  32. 32.Which of these numbers is divisible by 6? (a) 4615 (b) 4620 (c) 4630 (d) 4624
  33. 33.What is the smallest number you must add to 354 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 7
  34. 34.What is the smallest number you must add to 244 to make it divisible by 11? (a) 10 (b) 9 (c) 2 (d) 8
  35. 35.Which of these numbers is divisible by 4? (a) 1873 (b) 1882 (c) 1876 (d) 1878
  36. 36.Which of these numbers is divisible by 6? (a) 1818 (b) 1825 (c) 1813 (d) 1821
  37. 37.What is the smallest number you must add to 142 to make it divisible by 10? (a) 8 (b) 9 (c) 2 (d) 7
  38. 38.What is the units digit of 7^58? (a) 3 (b) 7 (c) 9 (d) 1
  39. 39.Which of these numbers is divisible by 3? (a) 2569 (b) 2566 (c) 2563 (d) 2565
  40. 40.What is the units digit of 4^37? (a) 5 (b) 6 (c) 3 (d) 4

Answer key

  1. 2
  2. 1
  3. 3
  4. 4
  5. 11
  6. 471
  7. 1104
  8. 4
  9. 6
  10. 3
  11. 3
  12. 7
  13. 1
  14. 6
  15. 1
  16. 6
  17. 5
  18. 2
  19. 6
  20. 3
  21. 516
  22. 585
  23. 2
  24. 6
  25. 1419
  26. 7
  27. 3
  28. 3
  29. 912
  30. 6
  31. 5
  32. 4620
  33. 6
  34. 9
  35. 1876
  36. 1818
  37. 8
  38. 9
  39. 2565
  40. 4

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