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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 653 · 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 9? (a) 1459 (b) 1458 (c) 1469 (d) 1457
  2. 2.What is the remainder when 130 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
  3. 3.What is the smallest number you must add to 798 to make it divisible by 10? (a) 8 (b) 1 (c) 2 (d) 3
  4. 4.What is the units digit of 4^51? (a) 6 (b) 4 (c) 3 (d) 5
  5. 5.Which of these numbers is divisible by 6? (a) 450 (b) 446 (c) 459 (d) 451
  6. 6.What is the units digit of 7^55? (a) 3 (b) 9 (c) 1 (d) 7
  7. 7.What is the smallest number you must add to 272 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 2
  8. 8.What is the units digit of 7^53? (a) 7 (b) 9 (c) 1 (d) 3
  9. 9.What is the units digit of 8^28? (a) 8 (b) 2 (c) 6 (d) 4
  10. 10.What is the remainder when 199 is divided by 3? (a) 0 (b) 2 (c) 1
  11. 11.What is the smallest number you must add to 91 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  12. 12.What is the remainder when 151 is divided by 11? (a) 9 (b) 3 (c) 8 (d) 7
  13. 13.What is the remainder when 483 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
  14. 14.What is the smallest number you must add to 686 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
  15. 15.What is the smallest number you must add to 191 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
  16. 16.What is the remainder when 191 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
  17. 17.What is the remainder when 548 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 1
  18. 18.What is the remainder when 214 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
  19. 19.Which of these numbers is divisible by 11? (a) 2735 (b) 2746 (c) 2728 (d) 2724
  20. 20.What is the remainder when 97 is divided by 8? (a) 1 (b) 7 (c) 2 (d) 0
  21. 21.What is the smallest number you must add to 182 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
  22. 22.What is the units digit of 7^40? (a) 7 (b) 1 (c) 3 (d) 9
  23. 23.What is the remainder when 364 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
  24. 24.Which of these numbers is divisible by 3? (a) 2660 (b) 2663 (c) 2665 (d) 2661
  25. 25.Which of these numbers is divisible by 9? (a) 7958 (b) 7979 (c) 7965 (d) 7967
  26. 26.What is the remainder when 224 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
  27. 27.What is the units digit of 2^23? (a) 6 (b) 8 (c) 4 (d) 2
  28. 28.What is the units digit of 7^30? (a) 3 (b) 7 (c) 9 (d) 1
  29. 29.What is the remainder when 174 is divided by 11? (a) 2 (b) 8 (c) 10 (d) 9
  30. 30.Which of these numbers is divisible by 11? (a) 7246 (b) 7233 (c) 7238 (d) 7259
  31. 31.What is the remainder when 255 is divided by 8? (a) 1 (b) 6 (c) 5 (d) 7
  32. 32.What is the smallest number you must add to 257 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  33. 33.Which of these numbers is divisible by 6? (a) 5314 (b) 5321 (c) 5316 (d) 5324
  34. 34.What is the units digit of 7^38? (a) 7 (b) 9 (c) 3 (d) 1
  35. 35.What is the remainder when 293 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  36. 36.What is the smallest number you must add to 719 to make it divisible by 11? (a) 4 (b) 8 (c) 7 (d) 6
  37. 37.What is the units digit of 9^44? (a) 1 (b) 0 (c) 9 (d) 2
  38. 38.What is the smallest number you must add to 123 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
  39. 39.What is the remainder when 151 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  40. 40.Which of these numbers is divisible by 8? (a) 3504 (b) 3500 (c) 3508 (d) 3516

Answer key

  1. 1458
  2. 2
  3. 2
  4. 4
  5. 450
  6. 3
  7. 8
  8. 7
  9. 6
  10. 1
  11. 1
  12. 8
  13. 3
  14. 10
  15. 5
  16. 5
  17. 2
  18. 4
  19. 2728
  20. 1
  21. 4
  22. 1
  23. 4
  24. 2661
  25. 7965
  26. 4
  27. 8
  28. 9
  29. 9
  30. 7238
  31. 7
  32. 3
  33. 5316
  34. 9
  35. 1
  36. 7
  37. 1
  38. 3
  39. 3
  40. 3504

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