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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 169 · 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 units digit of 7^18? (a) 1 (b) 9 (c) 3 (d) 7
  2. 2.What is the units digit of 8^60? (a) 8 (b) 4 (c) 2 (d) 6
  3. 3.What is the smallest number you must add to 412 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
  4. 4.What is the remainder when 323 is divided by 10? (a) 2 (b) 7 (c) 3 (d) 4
  5. 5.What is the units digit of 3^39? (a) 7 (b) 1 (c) 3 (d) 9
  6. 6.What is the smallest number you must add to 430 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 2
  7. 7.What is the remainder when 298 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  8. 8.Which of these numbers is divisible by 8? (a) 4849 (b) 4842 (c) 4848 (d) 4857
  9. 9.What is the remainder when 206 is divided by 8? (a) 6 (b) 7 (c) 5 (d) 2
  10. 10.What is the smallest number you must add to 725 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 4
  11. 11.What is the smallest number you must add to 185 to make it divisible by 3? (a) 2 (b) 1
  12. 12.What is the remainder when 483 is divided by 9? (a) 3 (b) 6 (c) 7 (d) 5
  13. 13.Which of these numbers is divisible by 11? (a) 450 (b) 467 (c) 451 (d) 457
  14. 14.What is the remainder when 928 is divided by 12? (a) 3 (b) 5 (c) 4 (d) 8
  15. 15.What is the smallest number you must add to 633 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
  16. 16.What is the remainder when 265 is divided by 11? (a) 2 (b) 0 (c) 1 (d) 10
  17. 17.What is the units digit of 8^16? (a) 4 (b) 8 (c) 6 (d) 2
  18. 18.Which of these numbers is divisible by 3? (a) 736 (b) 730 (c) 732 (d) 734
  19. 19.Which of these numbers is divisible by 8? (a) 5490 (b) 5503 (c) 5487 (d) 5488
  20. 20.What is the units digit of 3^47? (a) 3 (b) 7 (c) 9 (d) 1
  21. 21.Which of these numbers is divisible by 6? (a) 1833 (b) 1841 (c) 1830 (d) 1825
  22. 22.What is the remainder when 301 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
  23. 23.What is the units digit of 9^35? (a) 8 (b) 7 (c) 9 (d) 1
  24. 24.What is the smallest number you must add to 483 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
  25. 25.What is the remainder when 417 is divided by 12? (a) 3 (b) 9 (c) 8 (d) 10
  26. 26.What is the smallest number you must add to 182 to make it divisible by 3? (a) 1 (b) 2
  27. 27.What is the smallest number you must add to 124 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  28. 28.What is the remainder when 470 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
  29. 29.What is the units digit of 9^21? (a) 9 (b) 8 (c) 7 (d) 1
  30. 30.What is the remainder when 221 is divided by 3? (a) 1 (b) 2 (c) 0
  31. 31.Which of these numbers is divisible by 9? (a) 3164 (b) 3157 (c) 3150 (d) 3149
  32. 32.What is the smallest number you must add to 442 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
  33. 33.What is the smallest number you must add to 97 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
  34. 34.What is the smallest number you must add to 105 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
  35. 35.What is the remainder when 549 is divided by 10? (a) 9 (b) 7 (c) 8 (d) 1
  36. 36.Which of these numbers is divisible by 9? (a) 6061 (b) 6073 (c) 6066 (d) 6080
  37. 37.Which of these numbers is divisible by 6? (a) 2278 (b) 2290 (c) 2282 (d) 2280
  38. 38.What is the smallest number you must add to 34 to make it divisible by 3? (a) 2 (b) 1
  39. 39.What is the units digit of 9^51? (a) 7 (b) 1 (c) 8 (d) 9
  40. 40.Which of these numbers is divisible by 3? (a) 2012 (b) 2015 (c) 2010 (d) 2008

Answer key

  1. 9
  2. 6
  3. 2
  4. 3
  5. 7
  6. 2
  7. 2
  8. 4848
  9. 6
  10. 4
  11. 1
  12. 6
  13. 451
  14. 4
  15. 7
  16. 1
  17. 6
  18. 732
  19. 5488
  20. 7
  21. 1830
  22. 1
  23. 9
  24. 3
  25. 9
  26. 1
  27. 1
  28. 2
  29. 9
  30. 2
  31. 3150
  32. 8
  33. 5
  34. 7
  35. 9
  36. 6066
  37. 2280
  38. 2
  39. 9
  40. 2010

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