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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 802 · 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 smallest number you must add to 445 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
  2. 2.Which of these numbers is divisible by 4? (a) 2870 (b) 2873 (c) 2866 (d) 2868
  3. 3.What is the smallest number you must add to 208 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  4. 4.Which of these numbers is divisible by 9? (a) 4847 (b) 4837 (c) 4858 (d) 4842
  5. 5.What is the smallest number you must add to 136 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  6. 6.What is the remainder when 427 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
  7. 7.What is the remainder when 388 is divided by 11? (a) 4 (b) 8 (c) 2 (d) 3
  8. 8.Which of these numbers is divisible by 3? (a) 2534 (b) 2529 (c) 2530 (d) 2528
  9. 9.Which of these numbers is divisible by 4? (a) 876 (b) 877 (c) 875 (d) 881
  10. 10.Which of these numbers is divisible by 9? (a) 593 (b) 608 (c) 598 (d) 594
  11. 11.What is the smallest number you must add to 95 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
  12. 12.What is the remainder when 92 is divided by 3? (a) 1 (b) 0 (c) 2
  13. 13.What is the units digit of 3^21? (a) 9 (b) 1 (c) 3 (d) 7
  14. 14.Which of these numbers is divisible by 8? (a) 5883 (b) 5872 (c) 5867 (d) 5877
  15. 15.What is the smallest number you must add to 161 to make it divisible by 10? (a) 1 (b) 8 (c) 9 (d) 7
  16. 16.What is the smallest number you must add to 68 to make it divisible by 3? (a) 1 (b) 2
  17. 17.What is the units digit of 3^38? (a) 3 (b) 9 (c) 1 (d) 7
  18. 18.What is the smallest number you must add to 184 to make it divisible by 3? (a) 2 (b) 1
  19. 19.What is the remainder when 170 is divided by 8? (a) 3 (b) 1 (c) 2 (d) 6
  20. 20.What is the smallest number you must add to 233 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  21. 21.What is the smallest number you must add to 56 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  22. 22.Which of these numbers is divisible by 8? (a) 6976 (b) 6972 (c) 6977 (d) 6985
  23. 23.What is the remainder when 694 is divided by 11? (a) 2 (b) 0 (c) 10 (d) 1
  24. 24.What is the units digit of 2^14? (a) 4 (b) 6 (c) 2 (d) 8
  25. 25.What is the smallest number you must add to 106 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
  26. 26.Which of these numbers is divisible by 8? (a) 1799 (b) 1803 (c) 1809 (d) 1800
  27. 27.What is the units digit of 9^45? (a) 7 (b) 8 (c) 1 (d) 9
  28. 28.Which of these numbers is divisible by 4? (a) 1104 (b) 1107 (c) 1109 (d) 1103
  29. 29.What is the remainder when 878 is divided by 11? (a) 8 (b) 10 (c) 2 (d) 9
  30. 30.Which of these numbers is divisible by 4? (a) 2118 (b) 2120 (c) 2121 (d) 2126
  31. 31.What is the smallest number you must add to 91 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
  32. 32.What is the smallest number you must add to 585 to make it divisible by 8? (a) 5 (b) 1 (c) 7 (d) 6
  33. 33.What is the units digit of 4^28? (a) 7 (b) 5 (c) 4 (d) 6
  34. 34.What is the smallest number you must add to 191 to make it divisible by 9? (a) 2 (b) 6 (c) 7 (d) 8
  35. 35.Which of these numbers is divisible by 6? (a) 2416 (b) 2418 (c) 2421 (d) 2429
  36. 36.Which of these numbers is divisible by 9? (a) 4619 (b) 4611 (c) 4600 (d) 4608
  37. 37.What is the units digit of 9^18? (a) 9 (b) 0 (c) 2 (d) 1
  38. 38.What is the units digit of 8^10? (a) 2 (b) 8 (c) 6 (d) 4
  39. 39.What is the smallest number you must add to 293 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  40. 40.Which of these numbers is divisible by 4? (a) 3520 (b) 3517 (c) 3527 (d) 3521

Answer key

  1. 5
  2. 2868
  3. 2
  4. 4842
  5. 4
  6. 1
  7. 3
  8. 2529
  9. 876
  10. 594
  11. 3
  12. 2
  13. 3
  14. 5872
  15. 9
  16. 1
  17. 9
  18. 2
  19. 2
  20. 3
  21. 4
  22. 6976
  23. 1
  24. 4
  25. 2
  26. 1800
  27. 9
  28. 1104
  29. 9
  30. 2120
  31. 5
  32. 7
  33. 6
  34. 7
  35. 2418
  36. 4608
  37. 1
  38. 4
  39. 3
  40. 3520

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