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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 678 · 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 9? (a) 879 (b) 866 (c) 885 (d) 873
  2. 2.What is the smallest number you must add to 92 to make it divisible by 7? (a) 5 (b) 1 (c) 6 (d) 4
  3. 3.What is the smallest number you must add to 187 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 5
  4. 4.Which of these numbers is divisible by 8? (a) 5614 (b) 5600 (c) 5598 (d) 5606
  5. 5.What is the smallest number you must add to 507 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
  6. 6.What is the remainder when 230 is divided by 3? (a) 0 (b) 1 (c) 2
  7. 7.What is the units digit of 9^22? (a) 2 (b) 9 (c) 1 (d) 0
  8. 8.Which of these numbers is divisible by 3? (a) 1756 (b) 1755 (c) 1760 (d) 1753
  9. 9.Which of these numbers is divisible by 6? (a) 1627 (b) 1637 (c) 1632 (d) 1643
  10. 10.What is the smallest number you must add to 254 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  11. 11.What is the remainder when 215 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 1
  12. 12.Which of these numbers is divisible by 9? (a) 5630 (b) 5616 (c) 5609 (d) 5621
  13. 13.What is the units digit of 4^57? (a) 4 (b) 5 (c) 6 (d) 3
  14. 14.What is the remainder when 136 is divided by 10? (a) 4 (b) 7 (c) 6 (d) 5
  15. 15.What is the smallest number you must add to 278 to make it divisible by 7? (a) 2 (b) 3 (c) 5 (d) 1
  16. 16.What is the remainder when 192 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
  17. 17.What is the smallest number you must add to 239 to make it divisible by 8? (a) 7 (b) 2 (c) 1 (d) 3
  18. 18.Which of these numbers is divisible by 3? (a) 210 (b) 211 (c) 214 (d) 208
  19. 19.What is the remainder when 374 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 3
  20. 20.What is the remainder when 171 is divided by 10? (a) 2 (b) 1 (c) 9 (d) 0
  21. 21.Which of these numbers is divisible by 9? (a) 4865 (b) 4852 (c) 4851 (d) 4845
  22. 22.What is the units digit of 3^14? (a) 9 (b) 1 (c) 7 (d) 3
  23. 23.What is the units digit of 4^13? (a) 3 (b) 6 (c) 4 (d) 5
  24. 24.What is the smallest number you must add to 567 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3
  25. 25.What is the smallest number you must add to 395 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 2
  26. 26.What is the units digit of 7^28? (a) 9 (b) 3 (c) 7 (d) 1
  27. 27.What is the smallest number you must add to 465 to make it divisible by 9? (a) 2 (b) 6 (c) 3 (d) 4
  28. 28.What is the remainder when 567 is divided by 10? (a) 3 (b) 8 (c) 6 (d) 7
  29. 29.What is the smallest number you must add to 688 to make it divisible by 9? (a) 7 (b) 4 (c) 5 (d) 6
  30. 30.What is the units digit of 9^20? (a) 1 (b) 9 (c) 0 (d) 2
  31. 31.Which of these numbers is divisible by 9? (a) 6475 (b) 6462 (c) 6454 (d) 6468
  32. 32.What is the units digit of 8^19? (a) 4 (b) 6 (c) 8 (d) 2
  33. 33.What is the smallest number you must add to 628 to make it divisible by 10? (a) 2 (b) 1 (c) 3 (d) 8
  34. 34.What is the units digit of 8^37? (a) 4 (b) 2 (c) 8 (d) 6
  35. 35.What is the units digit of 2^22? (a) 6 (b) 8 (c) 4 (d) 2
  36. 36.What is the smallest number you must add to 57 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  37. 37.Which of these numbers is divisible by 4? (a) 1059 (b) 1055 (c) 1061 (d) 1056
  38. 38.Which of these numbers is divisible by 9? (a) 670 (b) 655 (c) 657 (d) 664
  39. 39.What is the remainder when 391 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 0
  40. 40.What is the units digit of 8^42? (a) 2 (b) 4 (c) 6 (d) 8

Answer key

  1. 873
  2. 6
  3. 2
  4. 5600
  5. 5
  6. 2
  7. 1
  8. 1755
  9. 1632
  10. 2
  11. 5
  12. 5616
  13. 4
  14. 6
  15. 2
  16. 5
  17. 1
  18. 210
  19. 4
  20. 1
  21. 4851
  22. 9
  23. 4
  24. 1
  25. 1
  26. 1
  27. 3
  28. 7
  29. 5
  30. 1
  31. 6462
  32. 2
  33. 2
  34. 8
  35. 4
  36. 3
  37. 1056
  38. 657
  39. 1
  40. 4

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