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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 39 · 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 2^11? (a) 4 (b) 8 (c) 2 (d) 6
  2. 2.What is the remainder when 196 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
  3. 3.What is the units digit of 7^53? (a) 1 (b) 7 (c) 9 (d) 3
  4. 4.What is the smallest number you must add to 38 to make it divisible by 3? (a) 1 (b) 2
  5. 5.What is the remainder when 222 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
  6. 6.What is the smallest number you must add to 102 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  7. 7.What is the units digit of 7^21? (a) 3 (b) 9 (c) 1 (d) 7
  8. 8.What is the smallest number you must add to 193 to make it divisible by 11? (a) 5 (b) 6 (c) 7 (d) 4
  9. 9.Which of these numbers is divisible by 3? (a) 2354 (b) 2355 (c) 2359 (d) 2357
  10. 10.What is the units digit of 2^40? (a) 4 (b) 2 (c) 6 (d) 8
  11. 11.What is the remainder when 314 is divided by 10? (a) 6 (b) 3 (c) 5 (d) 4
  12. 12.What is the remainder when 339 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
  13. 13.What is the units digit of 8^44? (a) 8 (b) 2 (c) 6 (d) 4
  14. 14.What is the smallest number you must add to 218 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
  15. 15.What is the smallest number you must add to 586 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 2
  16. 16.What is the remainder when 485 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
  17. 17.What is the smallest number you must add to 412 to make it divisible by 12? (a) 9 (b) 4 (c) 8 (d) 7
  18. 18.What is the units digit of 9^17? (a) 1 (b) 9 (c) 7 (d) 8
  19. 19.What is the smallest number you must add to 307 to make it divisible by 11? (a) 1 (b) 3 (c) 2 (d) 10
  20. 20.Which of these numbers is divisible by 3? (a) 2305 (b) 2308 (c) 2303 (d) 2304
  21. 21.What is the units digit of 9^36? (a) 0 (b) 1 (c) 9 (d) 2
  22. 22.What is the units digit of 8^40? (a) 2 (b) 4 (c) 8 (d) 6
  23. 23.What is the units digit of 8^55? (a) 8 (b) 4 (c) 2 (d) 6
  24. 24.What is the remainder when 173 is divided by 3? (a) 1 (b) 2 (c) 0
  25. 25.What is the units digit of 7^13? (a) 3 (b) 1 (c) 9 (d) 7
  26. 26.What is the units digit of 8^49? (a) 2 (b) 6 (c) 4 (d) 8
  27. 27.What is the remainder when 421 is divided by 10? (a) 0 (b) 1 (c) 2 (d) 9
  28. 28.What is the units digit of 8^50? (a) 8 (b) 2 (c) 6 (d) 4
  29. 29.What is the units digit of 3^31? (a) 7 (b) 9 (c) 3 (d) 1
  30. 30.What is the remainder when 206 is divided by 3? (a) 0 (b) 2 (c) 1
  31. 31.What is the remainder when 83 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
  32. 32.What is the smallest number you must add to 523 to make it divisible by 9? (a) 8 (b) 6 (c) 1 (d) 7
  33. 33.What is the units digit of 9^19? (a) 8 (b) 7 (c) 1 (d) 9
  34. 34.What is the smallest number you must add to 413 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 2
  35. 35.What is the remainder when 224 is divided by 12? (a) 8 (b) 7 (c) 4 (d) 9
  36. 36.What is the units digit of 9^13? (a) 8 (b) 7 (c) 9 (d) 1
  37. 37.What is the remainder when 155 is divided by 12? (a) 11 (b) 1 (c) 9 (d) 10
  38. 38.What is the units digit of 4^23? (a) 4 (b) 5 (c) 3 (d) 6
  39. 39.What is the smallest number you must add to 315 to make it divisible by 12? (a) 3 (b) 10 (c) 8 (d) 9
  40. 40.What is the units digit of 2^14? (a) 2 (b) 8 (c) 6 (d) 4

Answer key

  1. 8
  2. 6
  3. 7
  4. 1
  5. 6
  6. 2
  7. 7
  8. 5
  9. 2355
  10. 6
  11. 4
  12. 3
  13. 6
  14. 6
  15. 6
  16. 2
  17. 8
  18. 9
  19. 1
  20. 2304
  21. 1
  22. 6
  23. 2
  24. 2
  25. 7
  26. 8
  27. 1
  28. 4
  29. 7
  30. 2
  31. 3
  32. 8
  33. 9
  34. 1
  35. 8
  36. 9
  37. 11
  38. 4
  39. 9
  40. 4

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