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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 70 · 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) 2868 (b) 2853 (c) 2848 (d) 2855
  2. 2.What is the units digit of 8^29? (a) 6 (b) 4 (c) 8 (d) 2
  3. 3.What is the smallest number you must add to 298 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  4. 4.What is the remainder when 345 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
  5. 5.What is the remainder when 295 is divided by 7? (a) 6 (b) 1 (c) 0 (d) 2
  6. 6.What is the remainder when 74 is divided by 3? (a) 0 (b) 2 (c) 1
  7. 7.What is the remainder when 273 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
  8. 8.What is the units digit of 8^59? (a) 2 (b) 4 (c) 8 (d) 6
  9. 9.What is the smallest number you must add to 313 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  10. 10.What is the remainder when 114 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  11. 11.What is the units digit of 7^25? (a) 7 (b) 1 (c) 3 (d) 9
  12. 12.Which of these numbers is divisible by 3? (a) 2036 (b) 2038 (c) 2034 (d) 2033
  13. 13.Which of these numbers is divisible by 6? (a) 4350 (b) 4345 (c) 4359 (d) 4354
  14. 14.What is the units digit of 3^18? (a) 3 (b) 7 (c) 9 (d) 1
  15. 15.What is the units digit of 3^49? (a) 7 (b) 9 (c) 1 (d) 3
  16. 16.What is the units digit of 3^15? (a) 9 (b) 3 (c) 1 (d) 7
  17. 17.What is the smallest number you must add to 253 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
  18. 18.What is the remainder when 589 is divided by 12? (a) 2 (b) 1 (c) 11 (d) 0
  19. 19.What is the smallest number you must add to 334 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  20. 20.Which of these numbers is divisible by 9? (a) 2654 (b) 2641 (c) 2659 (d) 2646
  21. 21.What is the units digit of 3^52? (a) 1 (b) 3 (c) 7 (d) 9
  22. 22.What is the smallest number you must add to 899 to make it divisible by 12? (a) 11 (b) 1 (c) 3 (d) 2
  23. 23.What is the smallest number you must add to 404 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
  24. 24.What is the units digit of 9^16? (a) 0 (b) 2 (c) 9 (d) 1
  25. 25.What is the smallest number you must add to 545 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 1
  26. 26.What is the units digit of 9^25? (a) 1 (b) 8 (c) 7 (d) 9
  27. 27.What is the units digit of 3^35? (a) 1 (b) 3 (c) 9 (d) 7
  28. 28.What is the smallest number you must add to 231 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  29. 29.What is the remainder when 441 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
  30. 30.What is the remainder when 159 is divided by 7? (a) 5 (b) 2 (c) 6 (d) 4
  31. 31.Which of these numbers is divisible by 3? (a) 229 (b) 231 (c) 235 (d) 232
  32. 32.What is the units digit of 9^31? (a) 7 (b) 1 (c) 8 (d) 9
  33. 33.Which of these numbers is divisible by 8? (a) 6032 (b) 6033 (c) 6030 (d) 6041
  34. 34.What is the units digit of 9^55? (a) 9 (b) 1 (c) 7 (d) 8
  35. 35.What is the remainder when 712 is divided by 12? (a) 8 (b) 5 (c) 3 (d) 4
  36. 36.Which of these numbers is divisible by 9? (a) 3943 (b) 3933 (c) 3928 (d) 3937
  37. 37.What is the remainder when 360 is divided by 7? (a) 4 (b) 2 (c) 3 (d) 5
  38. 38.What is the remainder when 292 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
  39. 39.Which of these numbers is divisible by 8? (a) 1975 (b) 1976 (c) 1991 (d) 1983
  40. 40.What is the units digit of 7^49? (a) 3 (b) 1 (c) 9 (d) 7

Answer key

  1. 2853
  2. 8
  3. 2
  4. 5
  5. 1
  6. 2
  7. 3
  8. 2
  9. 3
  10. 2
  11. 7
  12. 2034
  13. 4350
  14. 9
  15. 3
  16. 7
  17. 5
  18. 1
  19. 1
  20. 2646
  21. 1
  22. 1
  23. 4
  24. 1
  25. 7
  26. 9
  27. 7
  28. 4
  29. 3
  30. 5
  31. 231
  32. 9
  33. 6032
  34. 9
  35. 4
  36. 3933
  37. 3
  38. 4
  39. 1976
  40. 7

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