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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 811 · 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 remainder when 669 is divided by 10? (a) 9 (b) 1 (c) 7 (d) 8
  2. 2.What is the smallest number you must add to 197 to make it divisible by 9? (a) 3 (b) 8 (c) 2 (d) 1
  3. 3.Which of these numbers is divisible by 8? (a) 5601 (b) 5595 (c) 5600 (d) 5614
  4. 4.What is the units digit of 8^23? (a) 8 (b) 4 (c) 6 (d) 2
  5. 5.What is the remainder when 89 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
  6. 6.What is the smallest number you must add to 221 to make it divisible by 11? (a) 1 (b) 9 (c) 8 (d) 10
  7. 7.What is the smallest number you must add to 252 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
  8. 8.Which of these numbers is divisible by 3? (a) 1064 (b) 1059 (c) 1058 (d) 1061
  9. 9.Which of these numbers is divisible by 4? (a) 1244 (b) 1247 (c) 1249 (d) 1242
  10. 10.What is the remainder when 558 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
  11. 11.What is the remainder when 351 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
  12. 12.What is the smallest number you must add to 47 to make it divisible by 3? (a) 1 (b) 2
  13. 13.What is the smallest number you must add to 225 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
  14. 14.Which of these numbers is divisible by 4? (a) 1486 (b) 1482 (c) 1484 (d) 1489
  15. 15.What is the remainder when 753 is divided by 12? (a) 3 (b) 9 (c) 8 (d) 10
  16. 16.Which of these numbers is divisible by 4? (a) 807 (b) 802 (c) 797 (d) 800
  17. 17.Which of these numbers is divisible by 11? (a) 3250 (b) 3261 (c) 3256 (d) 3274
  18. 18.What is the remainder when 163 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  19. 19.What is the remainder when 308 is divided by 9? (a) 7 (b) 1 (c) 3 (d) 2
  20. 20.Which of these numbers is divisible by 4? (a) 681 (b) 675 (c) 677 (d) 676
  21. 21.What is the smallest number you must add to 347 to make it divisible by 10? (a) 4 (b) 2 (c) 3 (d) 7
  22. 22.What is the units digit of 3^10? (a) 1 (b) 3 (c) 7 (d) 9
  23. 23.What is the smallest number you must add to 457 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
  24. 24.What is the units digit of 3^39? (a) 3 (b) 1 (c) 7 (d) 9
  25. 25.Which of these numbers is divisible by 9? (a) 5588 (b) 5578 (c) 5591 (d) 5580
  26. 26.What is the units digit of 9^55? (a) 1 (b) 9 (c) 8 (d) 7
  27. 27.What is the smallest number you must add to 223 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  28. 28.What is the smallest number you must add to 300 to make it divisible by 11? (a) 7 (b) 8 (c) 9 (d) 3
  29. 29.Which of these numbers is divisible by 9? (a) 2659 (b) 2655 (c) 2653 (d) 2669
  30. 30.What is the smallest number you must add to 729 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
  31. 31.What is the smallest number you must add to 382 to make it divisible by 9? (a) 7 (b) 4 (c) 5 (d) 6
  32. 32.Which of these numbers is divisible by 6? (a) 2993 (b) 2997 (c) 2985 (d) 2988
  33. 33.What is the smallest number you must add to 567 to make it divisible by 8? (a) 1 (b) 3 (c) 7 (d) 2
  34. 34.What is the remainder when 287 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
  35. 35.What is the remainder when 388 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  36. 36.What is the remainder when 122 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
  37. 37.What is the units digit of 8^54? (a) 6 (b) 2 (c) 4 (d) 8
  38. 38.What is the smallest number you must add to 202 to make it divisible by 3? (a) 1 (b) 2
  39. 39.What is the remainder when 487 is divided by 8? (a) 6 (b) 1 (c) 5 (d) 7
  40. 40.Which of these numbers is divisible by 4? (a) 821 (b) 818 (c) 814 (d) 816

Answer key

  1. 9
  2. 1
  3. 5600
  4. 2
  5. 1
  6. 10
  7. 3
  8. 1059
  9. 1244
  10. 5
  11. 1
  12. 1
  13. 6
  14. 1484
  15. 9
  16. 800
  17. 3256
  18. 3
  19. 2
  20. 676
  21. 3
  22. 9
  23. 5
  24. 7
  25. 5580
  26. 9
  27. 2
  28. 8
  29. 2655
  30. 1
  31. 5
  32. 2988
  33. 1
  34. 5
  35. 3
  36. 2
  37. 4
  38. 2
  39. 7
  40. 816

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