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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 701 · 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 8^20? (a) 8 (b) 2 (c) 6 (d) 4
  2. 2.What is the units digit of 9^12? (a) 9 (b) 0 (c) 2 (d) 1
  3. 3.Which of these numbers is divisible by 9? (a) 4406 (b) 4418 (c) 4401 (d) 4396
  4. 4.Which of these numbers is divisible by 9? (a) 5240 (b) 5249 (c) 5259 (d) 5247
  5. 5.Which of these numbers is divisible by 3? (a) 861 (b) 866 (c) 860 (d) 862
  6. 6.What is the smallest number you must add to 62 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  7. 7.What is the units digit of 2^11? (a) 4 (b) 2 (c) 8 (d) 6
  8. 8.What is the units digit of 9^21? (a) 1 (b) 9 (c) 8 (d) 7
  9. 9.What is the smallest number you must add to 691 to make it divisible by 12? (a) 5 (b) 4 (c) 7 (d) 6
  10. 10.What is the units digit of 7^39? (a) 3 (b) 1 (c) 7 (d) 9
  11. 11.What is the smallest number you must add to 145 to make it divisible by 9? (a) 1 (b) 6 (c) 8 (d) 7
  12. 12.What is the smallest number you must add to 328 to make it divisible by 7? (a) 6 (b) 3 (c) 1 (d) 2
  13. 13.What is the smallest number you must add to 162 to make it divisible by 11? (a) 2 (b) 8 (c) 3 (d) 4
  14. 14.What is the remainder when 470 is divided by 12? (a) 2 (b) 10 (c) 3 (d) 1
  15. 15.What is the smallest number you must add to 848 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 1
  16. 16.What is the remainder when 537 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
  17. 17.What is the smallest number you must add to 148 to make it divisible by 3? (a) 2 (b) 1
  18. 18.What is the smallest number you must add to 329 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 5
  19. 19.What is the units digit of 3^51? (a) 7 (b) 9 (c) 3 (d) 1
  20. 20.What is the smallest number you must add to 136 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
  21. 21.What is the units digit of 7^41? (a) 1 (b) 3 (c) 9 (d) 7
  22. 22.What is the units digit of 3^49? (a) 7 (b) 3 (c) 9 (d) 1
  23. 23.What is the remainder when 151 is divided by 3? (a) 2 (b) 0 (c) 1
  24. 24.What is the remainder when 141 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
  25. 25.What is the smallest number you must add to 153 to make it divisible by 11? (a) 10 (b) 3 (c) 1 (d) 2
  26. 26.Which of these numbers is divisible by 3? (a) 672 (b) 673 (c) 671 (d) 677
  27. 27.What is the remainder when 190 is divided by 8? (a) 7 (b) 5 (c) 2 (d) 6
  28. 28.What is the smallest number you must add to 842 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 6
  29. 29.What is the units digit of 7^11? (a) 9 (b) 7 (c) 3 (d) 1
  30. 30.What is the remainder when 80 is divided by 3? (a) 1 (b) 0 (c) 2
  31. 31.What is the remainder when 406 is divided by 9? (a) 1 (b) 0 (c) 2 (d) 8
  32. 32.What is the units digit of 7^24? (a) 1 (b) 9 (c) 3 (d) 7
  33. 33.What is the remainder when 664 is divided by 11? (a) 3 (b) 4 (c) 5 (d) 7
  34. 34.What is the units digit of 4^22? (a) 5 (b) 4 (c) 6 (d) 7
  35. 35.What is the smallest number you must add to 151 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  36. 36.What is the smallest number you must add to 366 to make it divisible by 9? (a) 2 (b) 4 (c) 6 (d) 3
  37. 37.Which of these numbers is divisible by 4? (a) 993 (b) 985 (c) 990 (d) 988
  38. 38.What is the units digit of 2^23? (a) 4 (b) 6 (c) 8 (d) 2
  39. 39.What is the smallest number you must add to 277 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
  40. 40.What is the smallest number you must add to 335 to make it divisible by 8? (a) 1 (b) 7 (c) 2 (d) 3

Answer key

  1. 6
  2. 1
  3. 4401
  4. 5247
  5. 861
  6. 3
  7. 8
  8. 9
  9. 5
  10. 3
  11. 8
  12. 1
  13. 3
  14. 2
  15. 10
  16. 1
  17. 2
  18. 4
  19. 7
  20. 2
  21. 7
  22. 3
  23. 1
  24. 1
  25. 1
  26. 672
  27. 6
  28. 5
  29. 3
  30. 2
  31. 1
  32. 1
  33. 4
  34. 6
  35. 1
  36. 3
  37. 988
  38. 8
  39. 5
  40. 1

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