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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 27 · 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 2^10? (a) 8 (b) 2 (c) 4 (d) 6
  2. 2.What is the remainder when 390 is divided by 8? (a) 7 (b) 5 (c) 2 (d) 6
  3. 3.What is the smallest number you must add to 543 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
  4. 4.What is the remainder when 258 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
  5. 5.What is the smallest number you must add to 453 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 1
  6. 6.Which of these numbers is divisible by 6? (a) 3898 (b) 3894 (c) 3892 (d) 3902
  7. 7.What is the units digit of 4^17? (a) 3 (b) 5 (c) 4 (d) 6
  8. 8.What is the remainder when 377 is divided by 11? (a) 8 (b) 3 (c) 2 (d) 4
  9. 9.What is the smallest number you must add to 329 to make it divisible by 10? (a) 9 (b) 1 (c) 2 (d) 3
  10. 10.Which of these numbers is divisible by 9? (a) 8022 (b) 8033 (c) 8016 (d) 8019
  11. 11.What is the remainder when 403 is divided by 10? (a) 2 (b) 4 (c) 7 (d) 3
  12. 12.What is the smallest number you must add to 307 to make it divisible by 7? (a) 3 (b) 1 (c) 2 (d) 6
  13. 13.Which of these numbers is divisible by 3? (a) 2527 (b) 2529 (c) 2530 (d) 2533
  14. 14.What is the units digit of 9^17? (a) 7 (b) 9 (c) 1 (d) 8
  15. 15.What is the remainder when 837 is divided by 11? (a) 0 (b) 10 (c) 2 (d) 1
  16. 16.Which of these numbers is divisible by 11? (a) 956 (b) 959 (c) 940 (d) 946
  17. 17.What is the smallest number you must add to 167 to make it divisible by 12? (a) 3 (b) 11 (c) 1 (d) 2
  18. 18.What is the remainder when 557 is divided by 8? (a) 6 (b) 5 (c) 4 (d) 3
  19. 19.What is the units digit of 7^39? (a) 7 (b) 1 (c) 3 (d) 9
  20. 20.What is the smallest number you must add to 125 to make it divisible by 3? (a) 1 (b) 2
  21. 21.Which of these numbers is divisible by 6? (a) 5186 (b) 5177 (c) 5178 (d) 5180
  22. 22.What is the units digit of 3^54? (a) 1 (b) 7 (c) 9 (d) 3
  23. 23.What is the remainder when 746 is divided by 12? (a) 10 (b) 3 (c) 2 (d) 1
  24. 24.What is the smallest number you must add to 219 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  25. 25.What is the units digit of 9^56? (a) 9 (b) 0 (c) 2 (d) 1
  26. 26.What is the smallest number you must add to 177 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
  27. 27.What is the smallest number you must add to 509 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 5
  28. 28.Which of these numbers is divisible by 6? (a) 3285 (b) 3282 (c) 3290 (d) 3280
  29. 29.What is the remainder when 250 is divided by 8? (a) 3 (b) 6 (c) 2 (d) 1
  30. 30.Which of these numbers is divisible by 9? (a) 5768 (b) 5744 (c) 5759 (d) 5751
  31. 31.Which of these numbers is divisible by 3? (a) 440 (b) 435 (c) 434 (d) 437
  32. 32.Which of these numbers is divisible by 6? (a) 5341 (b) 5346 (c) 5348 (d) 5354
  33. 33.What is the smallest number you must add to 586 to make it divisible by 9? (a) 6 (b) 1 (c) 8 (d) 7
  34. 34.What is the smallest number you must add to 174 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  35. 35.What is the remainder when 505 is divided by 8? (a) 0 (b) 1 (c) 7 (d) 2
  36. 36.Which of these numbers is divisible by 9? (a) 6930 (b) 6924 (c) 6941 (d) 6932
  37. 37.What is the remainder when 547 is divided by 11? (a) 9 (b) 8 (c) 3 (d) 7
  38. 38.What is the smallest number you must add to 110 to make it divisible by 9? (a) 6 (b) 2 (c) 7 (d) 8
  39. 39.What is the units digit of 4^45? (a) 5 (b) 3 (c) 6 (d) 4
  40. 40.What is the smallest number you must add to 188 to make it divisible by 3? (a) 2 (b) 1

Answer key

  1. 4
  2. 6
  3. 6
  4. 2
  5. 2
  6. 3894
  7. 4
  8. 3
  9. 1
  10. 8019
  11. 3
  12. 1
  13. 2529
  14. 9
  15. 1
  16. 946
  17. 1
  18. 5
  19. 3
  20. 1
  21. 5178
  22. 9
  23. 2
  24. 1
  25. 1
  26. 3
  27. 2
  28. 3282
  29. 2
  30. 5751
  31. 435
  32. 5346
  33. 8
  34. 1
  35. 1
  36. 6930
  37. 8
  38. 7
  39. 4
  40. 1

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