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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 882 · 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 243 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
  2. 2.What is the remainder when 578 is divided by 9? (a) 7 (b) 2 (c) 1 (d) 3
  3. 3.Which of these numbers is divisible by 3? (a) 1720 (b) 1724 (c) 1719 (d) 1717
  4. 4.What is the remainder when 78 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
  5. 5.What is the smallest number you must add to 121 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
  6. 6.What is the remainder when 41 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  7. 7.What is the units digit of 9^45? (a) 9 (b) 1 (c) 7 (d) 8
  8. 8.What is the smallest number you must add to 135 to make it divisible by 7? (a) 4 (b) 2 (c) 6 (d) 5
  9. 9.Which of these numbers is divisible by 11? (a) 9080 (b) 9090 (c) 9075 (d) 9067
  10. 10.Which of these numbers is divisible by 11? (a) 4669 (b) 4645 (c) 4662 (d) 4653
  11. 11.Which of these numbers is divisible by 11? (a) 8670 (b) 8651 (c) 8657 (d) 8665
  12. 12.What is the remainder when 356 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
  13. 13.What is the smallest number you must add to 142 to make it divisible by 10? (a) 2 (b) 9 (c) 7 (d) 8
  14. 14.What is the smallest number you must add to 120 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
  15. 15.Which of these numbers is divisible by 4? (a) 405 (b) 415 (c) 411 (d) 408
  16. 16.What is the units digit of 4^32? (a) 7 (b) 6 (c) 4 (d) 5
  17. 17.What is the remainder when 106 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
  18. 18.Which of these numbers is divisible by 6? (a) 1724 (b) 1715 (c) 1720 (d) 1716
  19. 19.Which of these numbers is divisible by 11? (a) 8842 (b) 8833 (c) 8828 (d) 8846
  20. 20.What is the smallest number you must add to 386 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  21. 21.What is the units digit of 4^25? (a) 4 (b) 5 (c) 3 (d) 6
  22. 22.What is the remainder when 391 is divided by 8? (a) 5 (b) 1 (c) 7 (d) 6
  23. 23.What is the units digit of 7^25? (a) 3 (b) 7 (c) 1 (d) 9
  24. 24.What is the units digit of 9^10? (a) 1 (b) 9 (c) 2 (d) 0
  25. 25.What is the units digit of 7^38? (a) 1 (b) 3 (c) 9 (d) 7
  26. 26.Which of these numbers is divisible by 6? (a) 2489 (b) 2492 (c) 2497 (d) 2490
  27. 27.What is the smallest number you must add to 151 to make it divisible by 7? (a) 2 (b) 5 (c) 4 (d) 3
  28. 28.Which of these numbers is divisible by 3? (a) 2314 (b) 2312 (c) 2310 (d) 2309
  29. 29.What is the smallest number you must add to 299 to make it divisible by 10? (a) 9 (b) 2 (c) 3 (d) 1
  30. 30.What is the smallest number you must add to 199 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  31. 31.Which of these numbers is divisible by 8? (a) 3739 (b) 3757 (c) 3744 (d) 3746
  32. 32.What is the units digit of 7^32? (a) 1 (b) 7 (c) 9 (d) 3
  33. 33.What is the units digit of 7^37? (a) 1 (b) 3 (c) 7 (d) 9
  34. 34.What is the smallest number you must add to 202 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  35. 35.What is the smallest number you must add to 383 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  36. 36.What is the smallest number you must add to 506 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 3
  37. 37.What is the units digit of 3^22? (a) 9 (b) 1 (c) 3 (d) 7
  38. 38.What is the smallest number you must add to 500 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
  39. 39.What is the remainder when 416 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
  40. 40.What is the smallest number you must add to 160 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 4

Answer key

  1. 3
  2. 2
  3. 1719
  4. 2
  5. 5
  6. 1
  7. 9
  8. 5
  9. 9075
  10. 4653
  11. 8657
  12. 6
  13. 8
  14. 6
  15. 408
  16. 6
  17. 2
  18. 1716
  19. 8833
  20. 4
  21. 4
  22. 7
  23. 7
  24. 1
  25. 9
  26. 2490
  27. 3
  28. 2310
  29. 1
  30. 1
  31. 3744
  32. 1
  33. 7
  34. 3
  35. 2
  36. 4
  37. 9
  38. 4
  39. 6
  40. 2

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