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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 783 · 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 235 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  2. 2.Which of these numbers is divisible by 6? (a) 4434 (b) 4429 (c) 4435 (d) 4445
  3. 3.Which of these numbers is divisible by 4? (a) 1890 (b) 1885 (c) 1888 (d) 1893
  4. 4.Which of these numbers is divisible by 4? (a) 957 (b) 956 (c) 962 (d) 953
  5. 5.What is the remainder when 205 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  6. 6.What is the smallest number you must add to 535 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 3
  7. 7.What is the remainder when 406 is divided by 8? (a) 2 (b) 7 (c) 5 (d) 6
  8. 8.Which of these numbers is divisible by 4? (a) 987 (b) 988 (c) 994 (d) 991
  9. 9.What is the remainder when 493 is divided by 10? (a) 4 (b) 2 (c) 7 (d) 3
  10. 10.What is the smallest number you must add to 183 to make it divisible by 11? (a) 5 (b) 3 (c) 4 (d) 7
  11. 11.What is the units digit of 9^17? (a) 1 (b) 8 (c) 7 (d) 9
  12. 12.What is the smallest number you must add to 354 to make it divisible by 8? (a) 2 (b) 7 (c) 6 (d) 5
  13. 13.What is the remainder when 508 is divided by 10? (a) 2 (b) 7 (c) 9 (d) 8
  14. 14.What is the units digit of 9^56? (a) 2 (b) 9 (c) 1 (d) 0
  15. 15.What is the units digit of 7^31? (a) 7 (b) 9 (c) 1 (d) 3
  16. 16.What is the units digit of 2^37? (a) 4 (b) 6 (c) 8 (d) 2
  17. 17.What is the smallest number you must add to 410 to make it divisible by 8? (a) 2 (b) 6 (c) 5 (d) 7
  18. 18.What is the units digit of 8^15? (a) 8 (b) 6 (c) 2 (d) 4
  19. 19.What is the units digit of 7^32? (a) 9 (b) 7 (c) 3 (d) 1
  20. 20.What is the remainder when 601 is divided by 8? (a) 1 (b) 0 (c) 2 (d) 7
  21. 21.Which of these numbers is divisible by 6? (a) 660 (b) 656 (c) 668 (d) 665
  22. 22.What is the remainder when 129 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
  23. 23.What is the units digit of 7^19? (a) 1 (b) 9 (c) 3 (d) 7
  24. 24.What is the smallest number you must add to 256 to make it divisible by 11? (a) 3 (b) 9 (c) 8 (d) 7
  25. 25.What is the units digit of 3^12? (a) 7 (b) 9 (c) 3 (d) 1
  26. 26.Which of these numbers is divisible by 4? (a) 2450 (b) 2442 (c) 2446 (d) 2444
  27. 27.What is the remainder when 163 is divided by 9? (a) 1 (b) 8 (c) 2 (d) 0
  28. 28.Which of these numbers is divisible by 11? (a) 5360 (b) 5368 (c) 5372 (d) 5385
  29. 29.Which of these numbers is divisible by 4? (a) 1007 (b) 997 (c) 1003 (d) 1000
  30. 30.What is the smallest number you must add to 292 to make it divisible by 8? (a) 5 (b) 6 (c) 3 (d) 4
  31. 31.What is the smallest number you must add to 135 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
  32. 32.What is the units digit of 3^44? (a) 9 (b) 7 (c) 3 (d) 1
  33. 33.What is the units digit of 2^47? (a) 6 (b) 2 (c) 8 (d) 4
  34. 34.What is the units digit of 9^24? (a) 1 (b) 2 (c) 0 (d) 9
  35. 35.What is the smallest number you must add to 171 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  36. 36.What is the units digit of 3^55? (a) 7 (b) 9 (c) 3 (d) 1
  37. 37.What is the smallest number you must add to 243 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
  38. 38.Which of these numbers is divisible by 6? (a) 774 (b) 779 (c) 781 (d) 772
  39. 39.Which of these numbers is divisible by 3? (a) 145 (b) 152 (c) 147 (d) 148
  40. 40.What is the smallest number you must add to 419 to make it divisible by 8? (a) 6 (b) 3 (c) 5 (d) 4

Answer key

  1. 1
  2. 4434
  3. 1888
  4. 956
  5. 1
  6. 4
  7. 6
  8. 988
  9. 3
  10. 4
  11. 9
  12. 6
  13. 8
  14. 1
  15. 3
  16. 2
  17. 6
  18. 2
  19. 1
  20. 1
  21. 660
  22. 4
  23. 3
  24. 8
  25. 1
  26. 2444
  27. 1
  28. 5368
  29. 1000
  30. 4
  31. 5
  32. 1
  33. 8
  34. 1
  35. 4
  36. 7
  37. 5
  38. 774
  39. 147
  40. 5

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