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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 548 · 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 4^47? (a) 6 (b) 5 (c) 4 (d) 3
  2. 2.What is the remainder when 144 is divided by 10? (a) 4 (b) 6 (c) 3 (d) 5
  3. 3.Which of these numbers is divisible by 4? (a) 462 (b) 469 (c) 464 (d) 467
  4. 4.What is the smallest number you must add to 243 to make it divisible by 7? (a) 5 (b) 2 (c) 1 (d) 3
  5. 5.What is the units digit of 3^30? (a) 9 (b) 3 (c) 1 (d) 7
  6. 6.What is the remainder when 414 is divided by 7? (a) 2 (b) 6 (c) 1 (d) 0
  7. 7.What is the remainder when 166 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 5
  8. 8.What is the remainder when 110 is divided by 3? (a) 0 (b) 2 (c) 1
  9. 9.What is the remainder when 561 is divided by 8? (a) 1 (b) 7 (c) 0 (d) 2
  10. 10.What is the units digit of 7^12? (a) 7 (b) 3 (c) 9 (d) 1
  11. 11.What is the units digit of 2^10? (a) 8 (b) 4 (c) 6 (d) 2
  12. 12.What is the units digit of 2^37? (a) 4 (b) 2 (c) 6 (d) 8
  13. 13.What is the remainder when 802 is divided by 11? (a) 9 (b) 8 (c) 10 (d) 1
  14. 14.Which of these numbers is divisible by 6? (a) 2772 (b) 2771 (c) 2781 (d) 2774
  15. 15.What is the smallest number you must add to 485 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
  16. 16.What is the units digit of 9^26? (a) 1 (b) 0 (c) 2 (d) 9
  17. 17.What is the smallest number you must add to 316 to make it divisible by 10? (a) 5 (b) 4 (c) 3 (d) 6
  18. 18.Which of these numbers is divisible by 4? (a) 705 (b) 703 (c) 709 (d) 704
  19. 19.What is the smallest number you must add to 513 to make it divisible by 7? (a) 2 (b) 6 (c) 5 (d) 4
  20. 20.What is the smallest number you must add to 110 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
  21. 21.Which of these numbers is divisible by 4? (a) 318 (b) 310 (c) 315 (d) 312
  22. 22.What is the units digit of 4^16? (a) 7 (b) 5 (c) 4 (d) 6
  23. 23.What is the smallest number you must add to 342 to make it divisible by 11? (a) 9 (b) 1 (c) 8 (d) 10
  24. 24.What is the remainder when 206 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
  25. 25.What is the units digit of 9^52? (a) 2 (b) 1 (c) 0 (d) 9
  26. 26.Which of these numbers is divisible by 9? (a) 4697 (b) 4699 (c) 4711 (d) 4698
  27. 27.What is the smallest number you must add to 634 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
  28. 28.Which of these numbers is divisible by 6? (a) 4530 (b) 4535 (c) 4538 (d) 4527
  29. 29.What is the smallest number you must add to 64 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
  30. 30.What is the smallest number you must add to 315 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
  31. 31.What is the remainder when 136 is divided by 3? (a) 1 (b) 2 (c) 0
  32. 32.Which of these numbers is divisible by 11? (a) 9295 (b) 9310 (c) 9294 (d) 9304
  33. 33.What is the smallest number you must add to 319 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  34. 34.Which of these numbers is divisible by 8? (a) 3640 (b) 3643 (c) 3635 (d) 3653
  35. 35.What is the smallest number you must add to 553 to make it divisible by 12? (a) 9 (b) 1 (c) 11 (d) 10
  36. 36.What is the units digit of 9^29? (a) 8 (b) 9 (c) 7 (d) 1
  37. 37.What is the remainder when 55 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  38. 38.What is the remainder when 208 is divided by 3? (a) 0 (b) 2 (c) 1
  39. 39.What is the remainder when 363 is divided by 6? (a) 5 (b) 2 (c) 3 (d) 4
  40. 40.Which of these numbers is divisible by 8? (a) 4920 (b) 4922 (c) 4917 (d) 4934

Answer key

  1. 4
  2. 4
  3. 464
  4. 2
  5. 9
  6. 1
  7. 4
  8. 2
  9. 1
  10. 1
  11. 4
  12. 2
  13. 10
  14. 2772
  15. 5
  16. 1
  17. 4
  18. 704
  19. 5
  20. 4
  21. 312
  22. 6
  23. 10
  24. 1
  25. 1
  26. 4698
  27. 6
  28. 4530
  29. 2
  30. 5
  31. 1
  32. 9295
  33. 1
  34. 3640
  35. 11
  36. 9
  37. 3
  38. 1
  39. 3
  40. 4920

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