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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 268 · 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 6? (a) 2450 (b) 2459 (c) 2448 (d) 2447
  2. 2.What is the smallest number you must add to 313 to make it divisible by 8? (a) 1 (b) 7 (c) 6 (d) 5
  3. 3.Which of these numbers is divisible by 4? (a) 759 (b) 760 (c) 767 (d) 761
  4. 4.What is the units digit of 9^16? (a) 1 (b) 9 (c) 2 (d) 0
  5. 5.What is the smallest number you must add to 250 to make it divisible by 12? (a) 3 (b) 2 (c) 1 (d) 10
  6. 6.What is the smallest number you must add to 274 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  7. 7.Which of these numbers is divisible by 3? (a) 1461 (b) 1459 (c) 1465 (d) 1463
  8. 8.What is the remainder when 130 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
  9. 9.Which of these numbers is divisible by 4? (a) 2661 (b) 2654 (c) 2656 (d) 2657
  10. 10.What is the remainder when 272 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
  11. 11.Which of these numbers is divisible by 11? (a) 2220 (b) 2208 (c) 2211 (d) 2231
  12. 12.What is the remainder when 164 is divided by 11? (a) 8 (b) 1 (c) 10 (d) 9
  13. 13.What is the remainder when 421 is divided by 7? (a) 2 (b) 6 (c) 0 (d) 1
  14. 14.Which of these numbers is divisible by 4? (a) 3549 (b) 3547 (c) 3554 (d) 3548
  15. 15.What is the smallest number you must add to 643 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5
  16. 16.What is the units digit of 8^31? (a) 6 (b) 8 (c) 4 (d) 2
  17. 17.What is the smallest number you must add to 231 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  18. 18.Which of these numbers is divisible by 9? (a) 975 (b) 998 (c) 981 (d) 982
  19. 19.What is the smallest number you must add to 215 to make it divisible by 12? (a) 3 (b) 11 (c) 1 (d) 2
  20. 20.Which of these numbers is divisible by 3? (a) 902 (b) 905 (c) 903 (d) 907
  21. 21.Which of these numbers is divisible by 4? (a) 1396 (b) 1402 (c) 1399 (d) 1395
  22. 22.What is the remainder when 784 is divided by 12? (a) 4 (b) 3 (c) 5 (d) 8
  23. 23.What is the smallest number you must add to 239 to make it divisible by 12? (a) 3 (b) 2 (c) 11 (d) 1
  24. 24.What is the remainder when 34 is divided by 3? (a) 0 (b) 2 (c) 1
  25. 25.What is the units digit of 3^51? (a) 1 (b) 7 (c) 3 (d) 9
  26. 26.What is the smallest number you must add to 236 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  27. 27.What is the smallest number you must add to 242 to make it divisible by 9? (a) 8 (b) 3 (c) 1 (d) 2
  28. 28.Which of these numbers is divisible by 9? (a) 1907 (b) 1898 (c) 1888 (d) 1890
  29. 29.Which of these numbers is divisible by 9? (a) 5589 (b) 5582 (c) 5590 (d) 5599
  30. 30.What is the remainder when 67 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
  31. 31.What is the remainder when 103 is divided by 7? (a) 6 (b) 5 (c) 2 (d) 4
  32. 32.What is the smallest number you must add to 757 to make it divisible by 10? (a) 3 (b) 2 (c) 7 (d) 4
  33. 33.What is the units digit of 8^48? (a) 8 (b) 2 (c) 6 (d) 4
  34. 34.What is the remainder when 454 is divided by 11? (a) 4 (b) 8 (c) 2 (d) 3
  35. 35.Which of these numbers is divisible by 9? (a) 7779 (b) 7785 (c) 7788 (d) 7797
  36. 36.What is the smallest number you must add to 165 to make it divisible by 12? (a) 9 (b) 3 (c) 4 (d) 2
  37. 37.What is the remainder when 161 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
  38. 38.Which of these numbers is divisible by 9? (a) 2122 (b) 2105 (c) 2106 (d) 2114
  39. 39.What is the units digit of 9^19? (a) 8 (b) 1 (c) 7 (d) 9
  40. 40.What is the smallest number you must add to 440 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4

Answer key

  1. 2448
  2. 7
  3. 760
  4. 1
  5. 2
  6. 2
  7. 1461
  8. 4
  9. 2656
  10. 2
  11. 2211
  12. 10
  13. 1
  14. 3548
  15. 5
  16. 2
  17. 4
  18. 981
  19. 1
  20. 903
  21. 1396
  22. 4
  23. 1
  24. 1
  25. 7
  26. 4
  27. 1
  28. 1890
  29. 5589
  30. 1
  31. 5
  32. 3
  33. 6
  34. 3
  35. 7785
  36. 3
  37. 8
  38. 2106
  39. 9
  40. 4

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