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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 906 · 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 remainder when 298 is divided by 12? (a) 2 (b) 11 (c) 10 (d) 9
  2. 2.What is the remainder when 329 is divided by 11? (a) 8 (b) 1 (c) 10 (d) 9
  3. 3.What is the smallest number you must add to 237 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  4. 4.Which of these numbers is divisible by 3? (a) 2183 (b) 2179 (c) 2176 (d) 2178
  5. 5.What is the smallest number you must add to 558 to make it divisible by 7? (a) 5 (b) 1 (c) 2 (d) 3
  6. 6.What is the smallest number you must add to 203 to make it divisible by 3? (a) 1 (b) 2
  7. 7.What is the smallest number you must add to 371 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  8. 8.What is the smallest number you must add to 78 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  9. 9.Which of these numbers is divisible by 9? (a) 6159 (b) 6176 (c) 6165 (d) 6166
  10. 10.What is the smallest number you must add to 86 to make it divisible by 7? (a) 6 (b) 2 (c) 4 (d) 5
  11. 11.What is the units digit of 8^23? (a) 2 (b) 6 (c) 4 (d) 8
  12. 12.What is the units digit of 2^53? (a) 6 (b) 4 (c) 2 (d) 8
  13. 13.What is the remainder when 489 is divided by 8? (a) 1 (b) 7 (c) 2 (d) 0
  14. 14.What is the units digit of 7^16? (a) 1 (b) 7 (c) 3 (d) 9
  15. 15.What is the remainder when 325 is divided by 11? (a) 8 (b) 5 (c) 7 (d) 6
  16. 16.What is the smallest number you must add to 570 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) 9163 (b) 9167 (c) 9153 (d) 9175
  18. 18.Which of these numbers is divisible by 11? (a) 6201 (b) 6182 (c) 6181 (d) 6192
  19. 19.What is the remainder when 517 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 1
  20. 20.What is the smallest number you must add to 562 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 2
  21. 21.What is the remainder when 280 is divided by 11? (a) 5 (b) 6 (c) 4 (d) 7
  22. 22.Which of these numbers is divisible by 3? (a) 613 (b) 610 (c) 608 (d) 609
  23. 23.Which of these numbers is divisible by 4? (a) 3175 (b) 3178 (c) 3176 (d) 3181
  24. 24.Which of these numbers is divisible by 8? (a) 3427 (b) 3420 (c) 3416 (d) 3414
  25. 25.What is the smallest number you must add to 155 to make it divisible by 7? (a) 6 (b) 5 (c) 1 (d) 4
  26. 26.What is the remainder when 147 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
  27. 27.What is the remainder when 446 is divided by 12? (a) 10 (b) 1 (c) 3 (d) 2
  28. 28.What is the smallest number you must add to 392 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
  29. 29.What is the remainder when 134 is divided by 3? (a) 1 (b) 0 (c) 2
  30. 30.Which of these numbers is divisible by 3? (a) 863 (b) 856 (c) 859 (d) 858
  31. 31.What is the units digit of 8^29? (a) 8 (b) 6 (c) 2 (d) 4
  32. 32.What is the smallest number you must add to 195 to make it divisible by 9? (a) 6 (b) 2 (c) 3 (d) 4
  33. 33.Which of these numbers is divisible by 9? (a) 2827 (b) 2852 (c) 2840 (d) 2835
  34. 34.What is the units digit of 7^27? (a) 3 (b) 7 (c) 1 (d) 9
  35. 35.What is the remainder when 122 is divided by 3? (a) 0 (b) 1 (c) 2
  36. 36.What is the remainder when 185 is divided by 8? (a) 1 (b) 0 (c) 7 (d) 2
  37. 37.What is the smallest number you must add to 143 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  38. 38.What is the units digit of 9^34? (a) 2 (b) 9 (c) 0 (d) 1
  39. 39.What is the units digit of 7^45? (a) 1 (b) 9 (c) 3 (d) 7
  40. 40.What is the smallest number you must add to 426 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6

Answer key

  1. 10
  2. 10
  3. 3
  4. 2178
  5. 2
  6. 1
  7. 4
  8. 2
  9. 6165
  10. 5
  11. 2
  12. 2
  13. 1
  14. 1
  15. 6
  16. 6
  17. 9163
  18. 6182
  19. 6
  20. 6
  21. 5
  22. 609
  23. 3176
  24. 3416
  25. 6
  26. 2
  27. 2
  28. 3
  29. 2
  30. 858
  31. 8
  32. 3
  33. 2835
  34. 3
  35. 2
  36. 1
  37. 1
  38. 1
  39. 7
  40. 1

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