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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 964 · 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 37 to make it divisible by 3? (a) 2 (b) 1
  2. 2.What is the units digit of 2^48? (a) 6 (b) 8 (c) 4 (d) 2
  3. 3.Which of these numbers is divisible by 3? (a) 571 (b) 570 (c) 569 (d) 574
  4. 4.What is the smallest number you must add to 178 to make it divisible by 9? (a) 3 (b) 1 (c) 2 (d) 7
  5. 5.What is the remainder when 488 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
  6. 6.What is the units digit of 3^31? (a) 7 (b) 9 (c) 1 (d) 3
  7. 7.Which of these numbers is divisible by 8? (a) 6765 (b) 6760 (c) 6755 (d) 6771
  8. 8.What is the remainder when 402 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 7
  9. 9.What is the smallest number you must add to 153 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  10. 10.What is the smallest number you must add to 273 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
  11. 11.What is the smallest number you must add to 142 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  12. 12.What is the smallest number you must add to 591 to make it divisible by 9? (a) 4 (b) 6 (c) 2 (d) 3
  13. 13.What is the remainder when 82 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  14. 14.What is the remainder when 285 is divided by 9? (a) 5 (b) 7 (c) 3 (d) 6
  15. 15.What is the remainder when 456 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
  16. 16.What is the smallest number you must add to 776 to make it divisible by 12? (a) 5 (b) 4 (c) 8 (d) 3
  17. 17.What is the smallest number you must add to 205 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  18. 18.What is the smallest number you must add to 202 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  19. 19.What is the remainder when 427 is divided by 9? (a) 3 (b) 4 (c) 5 (d) 6
  20. 20.What is the smallest number you must add to 208 to make it divisible by 3? (a) 2 (b) 1
  21. 21.What is the remainder when 140 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
  22. 22.What is the units digit of 9^31? (a) 1 (b) 9 (c) 7 (d) 8
  23. 23.What is the remainder when 142 is divided by 3? (a) 2 (b) 1 (c) 0
  24. 24.Which of these numbers is divisible by 6? (a) 2193 (b) 2196 (c) 2200 (d) 2203
  25. 25.What is the smallest number you must add to 267 to make it divisible by 11? (a) 8 (b) 7 (c) 9 (d) 3
  26. 26.Which of these numbers is divisible by 3? (a) 535 (b) 529 (c) 531 (d) 532
  27. 27.What is the smallest number you must add to 186 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  28. 28.What is the remainder when 511 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
  29. 29.What is the smallest number you must add to 271 to make it divisible by 11? (a) 5 (b) 4 (c) 7 (d) 3
  30. 30.What is the remainder when 403 is divided by 6? (a) 5 (b) 2 (c) 1 (d) 0
  31. 31.What is the units digit of 7^29? (a) 1 (b) 9 (c) 7 (d) 3
  32. 32.What is the units digit of 4^54? (a) 7 (b) 5 (c) 4 (d) 6
  33. 33.What is the remainder when 186 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
  34. 34.What is the remainder when 255 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  35. 35.What is the units digit of 3^35? (a) 9 (b) 7 (c) 3 (d) 1
  36. 36.What is the units digit of 3^44? (a) 3 (b) 9 (c) 7 (d) 1
  37. 37.Which of these numbers is divisible by 6? (a) 2638 (b) 2640 (c) 2643 (d) 2650
  38. 38.What is the units digit of 8^11? (a) 8 (b) 2 (c) 4 (d) 6
  39. 39.What is the smallest number you must add to 424 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
  40. 40.Which of these numbers is divisible by 8? (a) 6618 (b) 6630 (c) 6611 (d) 6616

Answer key

  1. 2
  2. 6
  3. 570
  4. 2
  5. 5
  6. 7
  7. 6760
  8. 6
  9. 2
  10. 7
  11. 3
  12. 3
  13. 2
  14. 6
  15. 1
  16. 4
  17. 3
  18. 3
  19. 4
  20. 2
  21. 4
  22. 9
  23. 1
  24. 2196
  25. 8
  26. 531
  27. 4
  28. 7
  29. 4
  30. 1
  31. 7
  32. 6
  33. 2
  34. 3
  35. 7
  36. 1
  37. 2640
  38. 2
  39. 2
  40. 6616

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