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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 304 · 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 230 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 2
  2. 2.Which of these numbers is divisible by 8? (a) 864 (b) 857 (c) 878 (d) 865
  3. 3.Which of these numbers is divisible by 6? (a) 3808 (b) 3811 (c) 3799 (d) 3804
  4. 4.What is the smallest number you must add to 187 to make it divisible by 9? (a) 2 (b) 7 (c) 3 (d) 1
  5. 5.Which of these numbers is divisible by 9? (a) 7516 (b) 7532 (c) 7534 (d) 7524
  6. 6.What is the smallest number you must add to 453 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
  7. 7.What is the smallest number you must add to 897 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
  8. 8.What is the units digit of 8^45? (a) 8 (b) 6 (c) 2 (d) 4
  9. 9.What is the smallest number you must add to 51 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  10. 10.What is the remainder when 167 is divided by 10? (a) 8 (b) 7 (c) 6 (d) 3
  11. 11.Which of these numbers is divisible by 3? (a) 2545 (b) 2542 (c) 2541 (d) 2539
  12. 12.What is the remainder when 306 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
  13. 13.Which of these numbers is divisible by 9? (a) 2943 (b) 2935 (c) 2953 (d) 2949
  14. 14.Which of these numbers is divisible by 8? (a) 1632 (b) 1633 (c) 1644 (d) 1631
  15. 15.What is the smallest number you must add to 351 to make it divisible by 12? (a) 10 (b) 3 (c) 8 (d) 9
  16. 16.What is the remainder when 125 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
  17. 17.What is the units digit of 4^14? (a) 4 (b) 7 (c) 5 (d) 6
  18. 18.What is the units digit of 3^37? (a) 1 (b) 3 (c) 9 (d) 7
  19. 19.Which of these numbers is divisible by 9? (a) 7488 (b) 7495 (c) 7502 (d) 7487
  20. 20.What is the smallest number you must add to 326 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
  21. 21.Which of these numbers is divisible by 11? (a) 5948 (b) 5951 (c) 5971 (d) 5958
  22. 22.What is the remainder when 527 is divided by 10? (a) 8 (b) 3 (c) 6 (d) 7
  23. 23.What is the smallest number you must add to 132 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
  24. 24.What is the smallest number you must add to 383 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 5
  25. 25.What is the remainder when 375 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 3
  26. 26.What is the remainder when 383 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
  27. 27.What is the smallest number you must add to 509 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 5
  28. 28.What is the units digit of 8^29? (a) 2 (b) 8 (c) 4 (d) 6
  29. 29.Which of these numbers is divisible by 8? (a) 1347 (b) 1359 (c) 1344 (d) 1339
  30. 30.What is the remainder when 347 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
  31. 31.What is the units digit of 9^56? (a) 2 (b) 1 (c) 0 (d) 9
  32. 32.What is the remainder when 105 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
  33. 33.What is the smallest number you must add to 415 to make it divisible by 12? (a) 7 (b) 4 (c) 6 (d) 5
  34. 34.What is the remainder when 405 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
  35. 35.What is the smallest number you must add to 250 to make it divisible by 11? (a) 4 (b) 8 (c) 3 (d) 2
  36. 36.Which of these numbers is divisible by 3? (a) 1977 (b) 1979 (c) 1982 (d) 1975
  37. 37.What is the smallest number you must add to 110 to make it divisible by 7? (a) 5 (b) 1 (c) 2 (d) 3
  38. 38.What is the units digit of 8^14? (a) 4 (b) 2 (c) 6 (d) 8
  39. 39.What is the remainder when 564 is divided by 7? (a) 5 (b) 3 (c) 4 (d) 6
  40. 40.Which of these numbers is divisible by 4? (a) 3444 (b) 3441 (c) 3451 (d) 3446

Answer key

  1. 10
  2. 864
  3. 3804
  4. 2
  5. 7524
  6. 3
  7. 3
  8. 8
  9. 1
  10. 7
  11. 2541
  12. 2
  13. 2943
  14. 1632
  15. 9
  16. 1
  17. 6
  18. 3
  19. 7488
  20. 4
  21. 5951
  22. 7
  23. 4
  24. 2
  25. 4
  26. 3
  27. 2
  28. 8
  29. 1344
  30. 2
  31. 1
  32. 1
  33. 5
  34. 5
  35. 3
  36. 1977
  37. 2
  38. 4
  39. 4
  40. 3444

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