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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 818 · 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 4? (a) 3199 (b) 3192 (c) 3195 (d) 3190
  2. 2.Which of these numbers is divisible by 3? (a) 1591 (b) 1595 (c) 1593 (d) 1598
  3. 3.What is the smallest number you must add to 177 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  4. 4.What is the units digit of 3^14? (a) 9 (b) 3 (c) 1 (d) 7
  5. 5.What is the remainder when 34 is divided by 3? (a) 0 (b) 2 (c) 1
  6. 6.What is the smallest number you must add to 61 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  7. 7.What is the units digit of 9^26? (a) 1 (b) 0 (c) 9 (d) 2
  8. 8.Which of these numbers is divisible by 4? (a) 650 (b) 653 (c) 648 (d) 647
  9. 9.What is the smallest number you must add to 95 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 4
  10. 10.What is the remainder when 531 is divided by 12? (a) 2 (b) 4 (c) 3 (d) 9
  11. 11.What is the units digit of 3^21? (a) 3 (b) 1 (c) 7 (d) 9
  12. 12.Which of these numbers is divisible by 9? (a) 2592 (b) 2590 (c) 2595 (d) 2602
  13. 13.What is the smallest number you must add to 411 to make it divisible by 12? (a) 10 (b) 8 (c) 9 (d) 3
  14. 14.Which of these numbers is divisible by 8? (a) 6624 (b) 6620 (c) 6627 (d) 6639
  15. 15.What is the smallest number you must add to 600 to make it divisible by 9? (a) 6 (b) 3 (c) 2 (d) 4
  16. 16.What is the units digit of 7^19? (a) 9 (b) 1 (c) 7 (d) 3
  17. 17.What is the units digit of 2^17? (a) 4 (b) 8 (c) 2 (d) 6
  18. 18.Which of these numbers is divisible by 6? (a) 333 (b) 327 (c) 320 (d) 324
  19. 19.What is the smallest number you must add to 493 to make it divisible by 9? (a) 1 (b) 7 (c) 3 (d) 2
  20. 20.What is the smallest number you must add to 750 to make it divisible by 12? (a) 8 (b) 6 (c) 7 (d) 5
  21. 21.What is the remainder when 188 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
  22. 22.Which of these numbers is divisible by 9? (a) 7440 (b) 7433 (c) 7425 (d) 7423
  23. 23.What is the smallest number you must add to 301 to make it divisible by 9? (a) 7 (b) 4 (c) 5 (d) 6
  24. 24.What is the smallest number you must add to 231 to make it divisible by 8? (a) 7 (b) 3 (c) 2 (d) 1
  25. 25.What is the remainder when 513 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
  26. 26.Which of these numbers is divisible by 9? (a) 4579 (b) 4566 (c) 4572 (d) 4582
  27. 27.What is the remainder when 165 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
  28. 28.What is the units digit of 3^12? (a) 9 (b) 1 (c) 7 (d) 3
  29. 29.What is the units digit of 3^46? (a) 3 (b) 9 (c) 1 (d) 7
  30. 30.Which of these numbers is divisible by 4? (a) 3443 (b) 3436 (c) 3437 (d) 3435
  31. 31.What is the smallest number you must add to 239 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
  32. 32.Which of these numbers is divisible by 3? (a) 1985 (b) 1990 (c) 1986 (d) 1988
  33. 33.What is the smallest number you must add to 520 to make it divisible by 9? (a) 1 (b) 3 (c) 2 (d) 7
  34. 34.What is the smallest number you must add to 234 to make it divisible by 8? (a) 5 (b) 2 (c) 7 (d) 6
  35. 35.What is the units digit of 7^16? (a) 7 (b) 9 (c) 3 (d) 1
  36. 36.What is the units digit of 9^18? (a) 1 (b) 0 (c) 9 (d) 2
  37. 37.What is the smallest number you must add to 206 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
  38. 38.Which of these numbers is divisible by 6? (a) 4964 (b) 4962 (c) 4959 (d) 4971
  39. 39.What is the units digit of 9^19? (a) 1 (b) 9 (c) 8 (d) 7
  40. 40.What is the smallest number you must add to 544 to make it divisible by 11? (a) 7 (b) 6 (c) 5 (d) 8

Answer key

  1. 3192
  2. 1593
  3. 3
  4. 9
  5. 1
  6. 4
  7. 1
  8. 648
  9. 4
  10. 3
  11. 3
  12. 2592
  13. 9
  14. 6624
  15. 3
  16. 3
  17. 2
  18. 324
  19. 2
  20. 6
  21. 6
  22. 7425
  23. 5
  24. 1
  25. 2
  26. 4572
  27. 1
  28. 1
  29. 9
  30. 3436
  31. 4
  32. 1986
  33. 2
  34. 6
  35. 1
  36. 1
  37. 4
  38. 4962
  39. 9
  40. 6

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