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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 958 · 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 109 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  2. 2.What is the units digit of 2^14? (a) 4 (b) 2 (c) 8 (d) 6
  3. 3.What is the remainder when 502 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
  4. 4.What is the remainder when 107 is divided by 10? (a) 6 (b) 8 (c) 7 (d) 3
  5. 5.What is the remainder when 735 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
  6. 6.What is the remainder when 81 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
  7. 7.Which of these numbers is divisible by 11? (a) 5639 (b) 5627 (c) 5632 (d) 5644
  8. 8.What is the smallest number you must add to 189 to make it divisible by 11? (a) 10 (b) 9 (c) 2 (d) 8
  9. 9.What is the smallest number you must add to 313 to make it divisible by 8? (a) 6 (b) 7 (c) 1 (d) 5
  10. 10.What is the remainder when 281 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
  11. 11.What is the units digit of 9^19? (a) 9 (b) 8 (c) 1 (d) 7
  12. 12.What is the remainder when 229 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 6
  13. 13.What is the units digit of 2^27? (a) 4 (b) 6 (c) 8 (d) 2
  14. 14.What is the smallest number you must add to 657 to make it divisible by 10? (a) 4 (b) 3 (c) 7 (d) 2
  15. 15.What is the remainder when 399 is divided by 11? (a) 3 (b) 8 (c) 2 (d) 4
  16. 16.What is the units digit of 9^43? (a) 8 (b) 9 (c) 7 (d) 1
  17. 17.What is the remainder when 88 is divided by 3? (a) 0 (b) 1 (c) 2
  18. 18.What is the remainder when 119 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  19. 19.Which of these numbers is divisible by 8? (a) 5390 (b) 5403 (c) 5393 (d) 5392
  20. 20.What is the smallest number you must add to 476 to make it divisible by 11? (a) 3 (b) 9 (c) 8 (d) 7
  21. 21.What is the smallest number you must add to 170 to make it divisible by 3? (a) 1 (b) 2
  22. 22.Which of these numbers is divisible by 3? (a) 1849 (b) 1853 (c) 1848 (d) 1846
  23. 23.What is the remainder when 628 is divided by 8? (a) 5 (b) 6 (c) 4 (d) 3
  24. 24.What is the remainder when 202 is divided by 3? (a) 2 (b) 1 (c) 0
  25. 25.What is the remainder when 112 is divided by 3? (a) 2 (b) 0 (c) 1
  26. 26.What is the smallest number you must add to 303 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 2
  27. 27.What is the remainder when 392 is divided by 12? (a) 4 (b) 8 (c) 9 (d) 7
  28. 28.What is the remainder when 241 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
  29. 29.What is the remainder when 56 is divided by 3? (a) 2 (b) 1 (c) 0
  30. 30.What is the smallest number you must add to 132 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
  31. 31.What is the smallest number you must add to 460 to make it divisible by 12? (a) 9 (b) 8 (c) 4 (d) 7
  32. 32.Which of these numbers is divisible by 3? (a) 2113 (b) 2112 (c) 2110 (d) 2117
  33. 33.Which of these numbers is divisible by 8? (a) 7042 (b) 7053 (c) 7059 (d) 7048
  34. 34.What is the smallest number you must add to 303 to make it divisible by 10? (a) 7 (b) 8 (c) 6 (d) 3
  35. 35.What is the remainder when 454 is divided by 7? (a) 4 (b) 6 (c) 1 (d) 5
  36. 36.What is the remainder when 68 is divided by 3? (a) 1 (b) 0 (c) 2
  37. 37.What is the smallest number you must add to 425 to make it divisible by 11? (a) 5 (b) 7 (c) 3 (d) 4
  38. 38.Which of these numbers is divisible by 3? (a) 993 (b) 997 (c) 991 (d) 995
  39. 39.What is the remainder when 525 is divided by 10? (a) 7 (b) 5 (c) 4 (d) 6
  40. 40.Which of these numbers is divisible by 8? (a) 2871 (b) 2881 (c) 2873 (d) 2872

Answer key

  1. 1
  2. 4
  3. 6
  4. 7
  5. 5
  6. 3
  7. 5632
  8. 9
  9. 7
  10. 1
  11. 9
  12. 5
  13. 8
  14. 3
  15. 3
  16. 9
  17. 1
  18. 4
  19. 5392
  20. 8
  21. 1
  22. 1848
  23. 4
  24. 1
  25. 1
  26. 3
  27. 8
  28. 1
  29. 2
  30. 3
  31. 8
  32. 2112
  33. 7048
  34. 7
  35. 6
  36. 2
  37. 4
  38. 993
  39. 5
  40. 2872

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