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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 446 · 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 218 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  2. 2.What is the remainder when 202 is divided by 3? (a) 1 (b) 0 (c) 2
  3. 3.What is the smallest number you must add to 463 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 4
  4. 4.What is the smallest number you must add to 235 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 1
  5. 5.What is the remainder when 113 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
  6. 6.What is the smallest number you must add to 233 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
  7. 7.What is the units digit of 9^38? (a) 2 (b) 1 (c) 9 (d) 0
  8. 8.Which of these numbers is divisible by 6? (a) 3743 (b) 3736 (c) 3738 (d) 3748
  9. 9.What is the units digit of 9^19? (a) 8 (b) 1 (c) 9 (d) 7
  10. 10.What is the smallest number you must add to 695 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4
  11. 11.What is the smallest number you must add to 125 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 2
  12. 12.What is the units digit of 3^24? (a) 7 (b) 1 (c) 3 (d) 9
  13. 13.What is the smallest number you must add to 149 to make it divisible by 3? (a) 2 (b) 1
  14. 14.What is the remainder when 201 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
  15. 15.What is the units digit of 3^21? (a) 7 (b) 1 (c) 9 (d) 3
  16. 16.What is the smallest number you must add to 201 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 3
  17. 17.Which of these numbers is divisible by 11? (a) 8850 (b) 8842 (c) 8831 (d) 8833
  18. 18.What is the smallest number you must add to 927 to make it divisible by 12? (a) 10 (b) 8 (c) 9 (d) 3
  19. 19.Which of these numbers is divisible by 9? (a) 6098 (b) 6088 (c) 6093 (d) 6107
  20. 20.What is the remainder when 109 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  21. 21.What is the smallest number you must add to 217 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  22. 22.What is the remainder when 432 is divided by 11? (a) 2 (b) 8 (c) 3 (d) 4
  23. 23.What is the smallest number you must add to 352 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 2
  24. 24.Which of these numbers is divisible by 9? (a) 7059 (b) 7040 (c) 7047 (d) 7051
  25. 25.Which of these numbers is divisible by 9? (a) 2091 (b) 2084 (c) 2098 (d) 2088
  26. 26.Which of these numbers is divisible by 4? (a) 3163 (b) 3164 (c) 3166 (d) 3169
  27. 27.Which of these numbers is divisible by 3? (a) 1585 (b) 1589 (c) 1587 (d) 1591
  28. 28.Which of these numbers is divisible by 8? (a) 2763 (b) 2770 (c) 2780 (d) 2768
  29. 29.What is the remainder when 253 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  30. 30.What is the remainder when 35 is divided by 3? (a) 0 (b) 1 (c) 2
  31. 31.What is the units digit of 9^34? (a) 0 (b) 1 (c) 2 (d) 9
  32. 32.Which of these numbers is divisible by 3? (a) 1474 (b) 1472 (c) 1477 (d) 1473
  33. 33.What is the units digit of 9^48? (a) 9 (b) 1 (c) 2 (d) 0
  34. 34.Which of these numbers is divisible by 4? (a) 469 (b) 468 (c) 466 (d) 475
  35. 35.Which of these numbers is divisible by 4? (a) 2237 (b) 2236 (c) 2241 (d) 2235
  36. 36.What is the smallest number you must add to 130 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
  37. 37.What is the smallest number you must add to 113 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
  38. 38.Which of these numbers is divisible by 11? (a) 2180 (b) 2178 (c) 2175 (d) 2197
  39. 39.What is the units digit of 2^43? (a) 4 (b) 6 (c) 2 (d) 8
  40. 40.What is the remainder when 349 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3

Answer key

  1. 2
  2. 1
  3. 5
  4. 5
  5. 3
  6. 2
  7. 1
  8. 3738
  9. 9
  10. 5
  11. 3
  12. 1
  13. 1
  14. 3
  15. 3
  16. 6
  17. 8833
  18. 9
  19. 6093
  20. 1
  21. 3
  22. 3
  23. 5
  24. 7047
  25. 2088
  26. 3164
  27. 1587
  28. 2768
  29. 1
  30. 2
  31. 1
  32. 1473
  33. 1
  34. 468
  35. 2236
  36. 2
  37. 6
  38. 2178
  39. 8
  40. 4

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