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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 275 · 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 273 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  2. 2.What is the smallest number you must add to 101 to make it divisible by 7? (a) 5 (b) 6 (c) 3 (d) 4
  3. 3.What is the smallest number you must add to 756 to make it divisible by 11? (a) 8 (b) 4 (c) 2 (d) 3
  4. 4.What is the remainder when 217 is divided by 3? (a) 0 (b) 1 (c) 2
  5. 5.Which of these numbers is divisible by 6? (a) 2667 (b) 2672 (c) 2660 (d) 2664
  6. 6.What is the smallest number you must add to 525 to make it divisible by 12? (a) 4 (b) 2 (c) 9 (d) 3
  7. 7.What is the smallest number you must add to 166 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  8. 8.What is the remainder when 63 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
  9. 9.What is the remainder when 694 is divided by 9? (a) 8 (b) 0 (c) 2 (d) 1
  10. 10.What is the smallest number you must add to 211 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  11. 11.Which of these numbers is divisible by 3? (a) 1364 (b) 1367 (c) 1365 (d) 1370
  12. 12.Which of these numbers is divisible by 11? (a) 4033 (b) 4015 (c) 4020 (d) 4007
  13. 13.Which of these numbers is divisible by 9? (a) 3974 (b) 3965 (c) 3969 (d) 3986
  14. 14.What is the units digit of 3^18? (a) 1 (b) 3 (c) 7 (d) 9
  15. 15.Which of these numbers is divisible by 8? (a) 6360 (b) 6375 (c) 6359 (d) 6362
  16. 16.Which of these numbers is divisible by 11? (a) 1181 (b) 1156 (c) 1174 (d) 1166
  17. 17.What is the smallest number you must add to 96 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  18. 18.Which of these numbers is divisible by 6? (a) 2515 (b) 2520 (c) 2527 (d) 2521
  19. 19.What is the remainder when 222 is divided by 10? (a) 3 (b) 8 (c) 2 (d) 1
  20. 20.Which of these numbers is divisible by 3? (a) 473 (b) 468 (c) 470 (d) 467
  21. 21.What is the units digit of 3^43? (a) 9 (b) 7 (c) 3 (d) 1
  22. 22.What is the units digit of 2^60? (a) 4 (b) 6 (c) 2 (d) 8
  23. 23.What is the units digit of 9^26? (a) 9 (b) 1 (c) 2 (d) 0
  24. 24.Which of these numbers is divisible by 3? (a) 2100 (b) 2105 (c) 2098 (d) 2101
  25. 25.Which of these numbers is divisible by 3? (a) 1786 (b) 1789 (c) 1785 (d) 1783
  26. 26.What is the remainder when 202 is divided by 3? (a) 0 (b) 2 (c) 1
  27. 27.Which of these numbers is divisible by 3? (a) 830 (b) 824 (c) 827 (d) 825
  28. 28.What is the remainder when 56 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
  29. 29.What is the smallest number you must add to 122 to make it divisible by 8? (a) 2 (b) 7 (c) 5 (d) 6
  30. 30.What is the smallest number you must add to 216 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  31. 31.What is the remainder when 385 is divided by 8? (a) 7 (b) 1 (c) 2 (d) 0
  32. 32.Which of these numbers is divisible by 4? (a) 3323 (b) 3318 (c) 3327 (d) 3320
  33. 33.What is the smallest number you must add to 321 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  34. 34.Which of these numbers is divisible by 3? (a) 1438 (b) 1440 (c) 1442 (d) 1445
  35. 35.Which of these numbers is divisible by 11? (a) 8909 (b) 8890 (c) 8917 (d) 8899
  36. 36.Which of these numbers is divisible by 3? (a) 1222 (b) 1226 (c) 1219 (d) 1221
  37. 37.Which of these numbers is divisible by 6? (a) 1045 (b) 1041 (c) 1036 (d) 1038
  38. 38.What is the smallest number you must add to 275 to make it divisible by 9? (a) 6 (b) 3 (c) 4 (d) 5
  39. 39.What is the remainder when 234 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
  40. 40.What is the remainder when 155 is divided by 3? (a) 0 (b) 1 (c) 2

Answer key

  1. 2
  2. 4
  3. 3
  4. 1
  5. 2664
  6. 3
  7. 4
  8. 3
  9. 1
  10. 1
  11. 1365
  12. 4015
  13. 3969
  14. 9
  15. 6360
  16. 1166
  17. 4
  18. 2520
  19. 2
  20. 468
  21. 7
  22. 6
  23. 1
  24. 2100
  25. 1785
  26. 1
  27. 825
  28. 1
  29. 6
  30. 4
  31. 1
  32. 3320
  33. 3
  34. 1440
  35. 8899
  36. 1221
  37. 1038
  38. 4
  39. 2
  40. 2

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