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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 894 · 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 609 to make it divisible by 8? (a) 1 (b) 7 (c) 6 (d) 5
  2. 2.Which of these numbers is divisible by 11? (a) 5725 (b) 5709 (c) 5719 (d) 5705
  3. 3.What is the remainder when 115 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  4. 4.What is the smallest number you must add to 120 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
  5. 5.What is the units digit of 7^24? (a) 7 (b) 9 (c) 3 (d) 1
  6. 6.What is the smallest number you must add to 355 to make it divisible by 9? (a) 6 (b) 7 (c) 4 (d) 5
  7. 7.What is the units digit of 8^17? (a) 6 (b) 4 (c) 8 (d) 2
  8. 8.Which of these numbers is divisible by 8? (a) 5468 (b) 5450 (c) 5456 (d) 5462
  9. 9.What is the smallest number you must add to 202 to make it divisible by 3? (a) 1 (b) 2
  10. 10.What is the smallest number you must add to 539 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4
  11. 11.What is the units digit of 7^42? (a) 3 (b) 1 (c) 9 (d) 7
  12. 12.What is the remainder when 407 is divided by 7? (a) 2 (b) 6 (c) 0 (d) 1
  13. 13.What is the remainder when 167 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
  14. 14.What is the units digit of 4^45? (a) 5 (b) 6 (c) 4 (d) 3
  15. 15.Which of these numbers is divisible by 11? (a) 6075 (b) 6072 (c) 6064 (d) 6087
  16. 16.Which of these numbers is divisible by 6? (a) 413 (b) 425 (c) 414 (d) 418
  17. 17.What is the units digit of 3^29? (a) 7 (b) 1 (c) 3 (d) 9
  18. 18.What is the units digit of 8^41? (a) 8 (b) 4 (c) 6 (d) 2
  19. 19.What is the remainder when 153 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
  20. 20.What is the units digit of 7^54? (a) 3 (b) 1 (c) 7 (d) 9
  21. 21.Which of these numbers is divisible by 8? (a) 4551 (b) 4534 (c) 4536 (d) 4543
  22. 22.Which of these numbers is divisible by 11? (a) 2888 (b) 2893 (c) 2895 (d) 2908
  23. 23.What is the units digit of 9^37? (a) 7 (b) 8 (c) 1 (d) 9
  24. 24.Which of these numbers is divisible by 9? (a) 6715 (b) 6728 (c) 6738 (d) 6723
  25. 25.What is the remainder when 137 is divided by 12? (a) 4 (b) 7 (c) 5 (d) 6
  26. 26.Which of these numbers is divisible by 8? (a) 6567 (b) 6568 (c) 6575 (d) 6578
  27. 27.What is the remainder when 220 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
  28. 28.What is the remainder when 662 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
  29. 29.What is the smallest number you must add to 236 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
  30. 30.Which of these numbers is divisible by 4? (a) 1641 (b) 1645 (c) 1650 (d) 1644
  31. 31.What is the smallest number you must add to 421 to make it divisible by 12? (a) 1 (b) 9 (c) 10 (d) 11
  32. 32.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
  33. 33.What is the smallest number you must add to 568 to make it divisible by 9? (a) 1 (b) 7 (c) 8 (d) 6
  34. 34.What is the smallest number you must add to 466 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 2
  35. 35.Which of these numbers is divisible by 4? (a) 1313 (b) 1309 (c) 1312 (d) 1319
  36. 36.What is the smallest number you must add to 441 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
  37. 37.Which of these numbers is divisible by 6? (a) 679 (b) 688 (c) 678 (d) 676
  38. 38.What is the smallest number you must add to 107 to make it divisible by 9? (a) 1 (b) 3 (c) 8 (d) 2
  39. 39.Which of these numbers is divisible by 4? (a) 2416 (b) 2415 (c) 2421 (d) 2417
  40. 40.What is the smallest number you must add to 728 to make it divisible by 9? (a) 1 (b) 8 (c) 2 (d) 3

Answer key

  1. 7
  2. 5709
  3. 3
  4. 6
  5. 1
  6. 5
  7. 8
  8. 5456
  9. 2
  10. 5
  11. 9
  12. 1
  13. 6
  14. 4
  15. 6072
  16. 414
  17. 3
  18. 8
  19. 3
  20. 9
  21. 4536
  22. 2893
  23. 9
  24. 6723
  25. 5
  26. 6568
  27. 4
  28. 2
  29. 4
  30. 1644
  31. 11
  32. 2
  33. 8
  34. 3
  35. 1312
  36. 3
  37. 678
  38. 1
  39. 2416
  40. 1

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