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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 161 · 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 11? (a) 8809 (b) 8797 (c) 8789 (d) 8779
  2. 2.Which of these numbers is divisible by 11? (a) 1105 (b) 1100 (c) 1117 (d) 1096
  3. 3.What is the remainder when 679 is divided by 11? (a) 9 (b) 3 (c) 8 (d) 7
  4. 4.What is the smallest number you must add to 119 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 1
  5. 5.What is the smallest number you must add to 491 to make it divisible by 9? (a) 4 (b) 6 (c) 5 (d) 3
  6. 6.Which of these numbers is divisible by 8? (a) 3541 (b) 3526 (c) 3528 (d) 3535
  7. 7.Which of these numbers is divisible by 9? (a) 2994 (b) 3003 (c) 2980 (d) 2988
  8. 8.Which of these numbers is divisible by 9? (a) 6199 (b) 6176 (c) 6183 (d) 6188
  9. 9.What is the smallest number you must add to 118 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 2
  10. 10.What is the units digit of 3^29? (a) 1 (b) 9 (c) 7 (d) 3
  11. 11.Which of these numbers is divisible by 9? (a) 2856 (b) 2848 (c) 2844 (d) 2840
  12. 12.What is the smallest number you must add to 140 to make it divisible by 9? (a) 3 (b) 6 (c) 4 (d) 5
  13. 13.What is the remainder when 724 is divided by 11? (a) 10 (b) 9 (c) 2 (d) 8
  14. 14.Which of these numbers is divisible by 6? (a) 1704 (b) 1714 (c) 1705 (d) 1699
  15. 15.What is the smallest number you must add to 677 to make it divisible by 10? (a) 4 (b) 7 (c) 2 (d) 3
  16. 16.Which of these numbers is divisible by 4? (a) 832 (b) 833 (c) 839 (d) 831
  17. 17.What is the smallest number you must add to 214 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
  18. 18.Which of these numbers is divisible by 9? (a) 2230 (b) 2235 (c) 2223 (d) 2216
  19. 19.What is the remainder when 175 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
  20. 20.What is the units digit of 4^18? (a) 6 (b) 5 (c) 7 (d) 4
  21. 21.What is the smallest number you must add to 91 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
  22. 22.What is the remainder when 394 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
  23. 23.What is the smallest number you must add to 240 to make it divisible by 7? (a) 5 (b) 2 (c) 6 (d) 4
  24. 24.What is the smallest number you must add to 839 to make it divisible by 11? (a) 8 (b) 7 (c) 9 (d) 3
  25. 25.Which of these numbers is divisible by 11? (a) 6768 (b) 6782 (c) 6765 (d) 6759
  26. 26.Which of these numbers is divisible by 11? (a) 7842 (b) 7853 (c) 7832 (d) 7831
  27. 27.What is the remainder when 550 is divided by 9? (a) 8 (b) 0 (c) 1 (d) 2
  28. 28.What is the smallest number you must add to 116 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
  29. 29.What is the remainder when 685 is divided by 11? (a) 3 (b) 2 (c) 8 (d) 4
  30. 30.What is the remainder when 508 is divided by 7? (a) 3 (b) 6 (c) 5 (d) 4
  31. 31.What is the units digit of 7^42? (a) 1 (b) 3 (c) 9 (d) 7
  32. 32.What is the units digit of 4^35? (a) 6 (b) 4 (c) 5 (d) 3
  33. 33.What is the remainder when 167 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
  34. 34.Which of these numbers is divisible by 4? (a) 2696 (b) 2693 (c) 2701 (d) 2697
  35. 35.What is the smallest number you must add to 399 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
  36. 36.What is the smallest number you must add to 271 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  37. 37.What is the units digit of 8^13? (a) 8 (b) 2 (c) 4 (d) 6
  38. 38.Which of these numbers is divisible by 8? (a) 6426 (b) 6438 (c) 6445 (d) 6432
  39. 39.What is the units digit of 4^15? (a) 3 (b) 4 (c) 6 (d) 5
  40. 40.What is the units digit of 7^11? (a) 3 (b) 7 (c) 1 (d) 9

Answer key

  1. 8789
  2. 1100
  3. 8
  4. 1
  5. 4
  6. 3528
  7. 2988
  8. 6183
  9. 2
  10. 3
  11. 2844
  12. 4
  13. 9
  14. 1704
  15. 3
  16. 832
  17. 6
  18. 2223
  19. 3
  20. 6
  21. 8
  22. 2
  23. 5
  24. 8
  25. 6765
  26. 7832
  27. 1
  28. 4
  29. 3
  30. 4
  31. 9
  32. 4
  33. 3
  34. 2696
  35. 6
  36. 1
  37. 8
  38. 6432
  39. 4
  40. 3

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