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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 698 · 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.Which of these numbers is divisible by 9? (a) 6179 (b) 6172 (c) 6161 (d) 6165
  2. 2.Which of these numbers is divisible by 11? (a) 8088 (b) 8085 (c) 8080 (d) 8102
  3. 3.What is the smallest number you must add to 137 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  4. 4.Which of these numbers is divisible by 6? (a) 930 (b) 940 (c) 929 (d) 931
  5. 5.What is the remainder when 76 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
  6. 6.What is the smallest number you must add to 218 to make it divisible by 8? (a) 2 (b) 7 (c) 5 (d) 6
  7. 7.Which of these numbers is divisible by 3? (a) 967 (b) 971 (c) 969 (d) 973
  8. 8.What is the smallest number you must add to 425 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 5
  9. 9.What is the remainder when 57 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  10. 10.What is the remainder when 497 is divided by 10? (a) 8 (b) 7 (c) 3 (d) 6
  11. 11.What is the units digit of 9^28? (a) 2 (b) 0 (c) 1 (d) 9
  12. 12.What is the remainder when 150 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
  13. 13.What is the smallest number you must add to 485 to make it divisible by 9? (a) 1 (b) 8 (c) 3 (d) 2
  14. 14.What is the smallest number you must add to 847 to make it divisible by 12? (a) 4 (b) 5 (c) 7 (d) 6
  15. 15.What is the remainder when 263 is divided by 10? (a) 4 (b) 2 (c) 3 (d) 7
  16. 16.Which of these numbers is divisible by 8? (a) 4494 (b) 4503 (c) 4488 (d) 4481
  17. 17.What is the units digit of 3^43? (a) 7 (b) 1 (c) 3 (d) 9
  18. 18.What is the remainder when 476 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
  19. 19.What is the smallest number you must add to 193 to make it divisible by 8? (a) 1 (b) 7 (c) 6 (d) 5
  20. 20.What is the units digit of 2^38? (a) 6 (b) 8 (c) 4 (d) 2
  21. 21.What is the remainder when 75 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  22. 22.What is the units digit of 4^57? (a) 5 (b) 4 (c) 6 (d) 3
  23. 23.What is the units digit of 3^27? (a) 1 (b) 9 (c) 3 (d) 7
  24. 24.What is the units digit of 4^23? (a) 3 (b) 4 (c) 5 (d) 6
  25. 25.What is the smallest number you must add to 626 to make it divisible by 9? (a) 6 (b) 4 (c) 5 (d) 3
  26. 26.What is the units digit of 9^56? (a) 1 (b) 9 (c) 0 (d) 2
  27. 27.What is the units digit of 2^50? (a) 6 (b) 8 (c) 2 (d) 4
  28. 28.What is the units digit of 9^58? (a) 2 (b) 0 (c) 9 (d) 1
  29. 29.What is the remainder when 306 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
  30. 30.Which of these numbers is divisible by 3? (a) 1991 (b) 1986 (c) 1987 (d) 1984
  31. 31.What is the smallest number you must add to 412 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
  32. 32.Which of these numbers is divisible by 4? (a) 2429 (b) 2424 (c) 2426 (d) 2422
  33. 33.What is the smallest number you must add to 412 to make it divisible by 9? (a) 2 (b) 1 (c) 3 (d) 7
  34. 34.What is the units digit of 4^54? (a) 5 (b) 4 (c) 7 (d) 6
  35. 35.What is the smallest number you must add to 443 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
  36. 36.What is the remainder when 535 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 3
  37. 37.What is the units digit of 4^44? (a) 6 (b) 5 (c) 4 (d) 7
  38. 38.What is the remainder when 214 is divided by 3? (a) 0 (b) 1 (c) 2
  39. 39.What is the units digit of 9^19? (a) 1 (b) 8 (c) 9 (d) 7
  40. 40.What is the remainder when 220 is divided by 12? (a) 8 (b) 3 (c) 4 (d) 5

Answer key

  1. 6165
  2. 8085
  3. 3
  4. 930
  5. 4
  6. 6
  7. 969
  8. 1
  9. 1
  10. 7
  11. 1
  12. 2
  13. 1
  14. 5
  15. 3
  16. 4488
  17. 7
  18. 2
  19. 7
  20. 4
  21. 3
  22. 4
  23. 7
  24. 4
  25. 4
  26. 1
  27. 4
  28. 1
  29. 2
  30. 1986
  31. 4
  32. 2424
  33. 2
  34. 6
  35. 5
  36. 3
  37. 6
  38. 1
  39. 9
  40. 4

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