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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 344 · 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 197 to make it divisible by 3? (a) 1 (b) 2
  2. 2.What is the units digit of 4^17? (a) 6 (b) 3 (c) 4 (d) 5
  3. 3.What is the smallest number you must add to 249 to make it divisible by 7? (a) 4 (b) 3 (c) 5 (d) 2
  4. 4.What is the remainder when 208 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
  5. 5.What is the smallest number you must add to 499 to make it divisible by 8? (a) 4 (b) 6 (c) 3 (d) 5
  6. 6.Which of these numbers is divisible by 4? (a) 525 (b) 522 (c) 520 (d) 519
  7. 7.What is the smallest number you must add to 593 to make it divisible by 8? (a) 6 (b) 7 (c) 1 (d) 5
  8. 8.Which of these numbers is divisible by 3? (a) 2288 (b) 2290 (c) 2286 (d) 2285
  9. 9.What is the smallest number you must add to 419 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
  10. 10.Which of these numbers is divisible by 4? (a) 661 (b) 665 (c) 660 (d) 659
  11. 11.What is the smallest number you must add to 171 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
  12. 12.What is the smallest number you must add to 273 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
  13. 13.What is the units digit of 9^34? (a) 2 (b) 0 (c) 9 (d) 1
  14. 14.What is the remainder when 127 is divided by 7? (a) 0 (b) 6 (c) 2 (d) 1
  15. 15.What is the smallest number you must add to 353 to make it divisible by 11? (a) 8 (b) 10 (c) 1 (d) 9
  16. 16.What is the remainder when 380 is divided by 12? (a) 9 (b) 8 (c) 7 (d) 4
  17. 17.Which of these numbers is divisible by 8? (a) 3128 (b) 3137 (c) 3133 (d) 3124
  18. 18.Which of these numbers is divisible by 9? (a) 3489 (b) 3466 (c) 3474 (d) 3480
  19. 19.What is the units digit of 7^11? (a) 1 (b) 3 (c) 9 (d) 7
  20. 20.What is the remainder when 68 is divided by 3? (a) 0 (b) 1 (c) 2
  21. 21.What is the smallest number you must add to 428 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
  22. 22.What is the smallest number you must add to 105 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  23. 23.What is the smallest number you must add to 618 to make it divisible by 11? (a) 2 (b) 9 (c) 8 (d) 10
  24. 24.Which of these numbers is divisible by 3? (a) 2190 (b) 2192 (c) 2194 (d) 2188
  25. 25.What is the remainder when 137 is divided by 3? (a) 2 (b) 1 (c) 0
  26. 26.What is the units digit of 7^16? (a) 7 (b) 3 (c) 9 (d) 1
  27. 27.Which of these numbers is divisible by 3? (a) 1133 (b) 1134 (c) 1135 (d) 1138
  28. 28.What is the remainder when 339 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
  29. 29.What is the remainder when 262 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 3
  30. 30.What is the units digit of 7^33? (a) 1 (b) 3 (c) 9 (d) 7
  31. 31.Which of these numbers is divisible by 11? (a) 9889 (b) 9910 (c) 9888 (d) 9891
  32. 32.Which of these numbers is divisible by 4? (a) 2335 (b) 2326 (c) 2328 (d) 2329
  33. 33.What is the smallest number you must add to 123 to make it divisible by 12? (a) 9 (b) 3 (c) 10 (d) 8
  34. 34.Which of these numbers is divisible by 4? (a) 1120 (b) 1122 (c) 1127 (d) 1119
  35. 35.Which of these numbers is divisible by 3? (a) 1885 (b) 1879 (c) 1881 (d) 1882
  36. 36.What is the smallest number you must add to 358 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
  37. 37.Which of these numbers is divisible by 9? (a) 1521 (b) 1523 (c) 1519 (d) 1531
  38. 38.What is the smallest number you must add to 340 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
  39. 39.What is the remainder when 699 is divided by 11? (a) 5 (b) 8 (c) 7 (d) 6
  40. 40.Which of these numbers is divisible by 4? (a) 2528 (b) 2525 (c) 2535 (d) 2529

Answer key

  1. 1
  2. 4
  3. 3
  4. 3
  5. 5
  6. 520
  7. 7
  8. 2286
  9. 5
  10. 660
  11. 3
  12. 3
  13. 1
  14. 1
  15. 10
  16. 8
  17. 3128
  18. 3474
  19. 3
  20. 2
  21. 4
  22. 3
  23. 9
  24. 2190
  25. 2
  26. 1
  27. 1134
  28. 3
  29. 3
  30. 7
  31. 9889
  32. 2328
  33. 9
  34. 1120
  35. 1881
  36. 6
  37. 1521
  38. 2
  39. 6
  40. 2528

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