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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 778 · 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 129 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  2. 2.What is the units digit of 7^17? (a) 9 (b) 3 (c) 7 (d) 1
  3. 3.Which of these numbers is divisible by 8? (a) 6226 (b) 6242 (c) 6232 (d) 6236
  4. 4.Which of these numbers is divisible by 8? (a) 1200 (b) 1206 (c) 1196 (d) 1210
  5. 5.What is the smallest number you must add to 307 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
  6. 6.Which of these numbers is divisible by 9? (a) 4177 (b) 4176 (c) 4186 (d) 4169
  7. 7.What is the remainder when 304 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
  8. 8.What is the smallest number you must add to 234 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
  9. 9.What is the smallest number you must add to 337 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
  10. 10.Which of these numbers is divisible by 11? (a) 1023 (b) 1020 (c) 1036 (d) 1033
  11. 11.What is the remainder when 853 is divided by 12? (a) 0 (b) 11 (c) 1 (d) 2
  12. 12.What is the units digit of 3^12? (a) 9 (b) 7 (c) 3 (d) 1
  13. 13.What is the remainder when 456 is divided by 9? (a) 7 (b) 6 (c) 3 (d) 5
  14. 14.Which of these numbers is divisible by 3? (a) 945 (b) 949 (c) 944 (d) 947
  15. 15.What is the smallest number you must add to 210 to make it divisible by 8? (a) 2 (b) 5 (c) 7 (d) 6
  16. 16.Which of these numbers is divisible by 3? (a) 1139 (b) 1141 (c) 1137 (d) 1135
  17. 17.What is the units digit of 8^48? (a) 4 (b) 8 (c) 6 (d) 2
  18. 18.What is the smallest number you must add to 103 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
  19. 19.Which of these numbers is divisible by 9? (a) 2868 (b) 2871 (c) 2875 (d) 2888
  20. 20.What is the remainder when 694 is divided by 10? (a) 5 (b) 6 (c) 3 (d) 4
  21. 21.Which of these numbers is divisible by 3? (a) 1210 (b) 1212 (c) 1213 (d) 1216
  22. 22.Which of these numbers is divisible by 9? (a) 3099 (b) 3111 (c) 3096 (d) 3095
  23. 23.Which of these numbers is divisible by 6? (a) 2387 (b) 2393 (c) 2395 (d) 2388
  24. 24.Which of these numbers is divisible by 11? (a) 6724 (b) 6713 (c) 6742 (d) 6721
  25. 25.Which of these numbers is divisible by 6? (a) 2479 (b) 2472 (c) 2473 (d) 2470
  26. 26.What is the smallest number you must add to 539 to make it divisible by 9? (a) 2 (b) 8 (c) 3 (d) 1
  27. 27.What is the remainder when 674 is divided by 10? (a) 3 (b) 5 (c) 6 (d) 4
  28. 28.Which of these numbers is divisible by 11? (a) 5044 (b) 5027 (c) 5037 (d) 5020
  29. 29.What is the remainder when 184 is divided by 3? (a) 1 (b) 0 (c) 2
  30. 30.What is the smallest number you must add to 58 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  31. 31.What is the units digit of 2^30? (a) 4 (b) 2 (c) 8 (d) 6
  32. 32.What is the remainder when 291 is divided by 9? (a) 6 (b) 2 (c) 4 (d) 3
  33. 33.What is the smallest number you must add to 573 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 2
  34. 34.What is the units digit of 8^12? (a) 2 (b) 6 (c) 4 (d) 8
  35. 35.What is the smallest number you must add to 157 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 1
  36. 36.What is the smallest number you must add to 322 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
  37. 37.Which of these numbers is divisible by 6? (a) 1459 (b) 1453 (c) 1452 (d) 1447
  38. 38.What is the remainder when 279 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  39. 39.Which of these numbers is divisible by 11? (a) 6272 (b) 6281 (c) 6300 (d) 6289
  40. 40.What is the smallest number you must add to 137 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3

Answer key

  1. 3
  2. 7
  3. 6232
  4. 1200
  5. 5
  6. 4176
  7. 4
  8. 6
  9. 3
  10. 1023
  11. 1
  12. 1
  13. 6
  14. 945
  15. 6
  16. 1137
  17. 6
  18. 5
  19. 2871
  20. 4
  21. 1212
  22. 3096
  23. 2388
  24. 6721
  25. 2472
  26. 1
  27. 4
  28. 5027
  29. 1
  30. 2
  31. 4
  32. 3
  33. 3
  34. 6
  35. 11
  36. 2
  37. 1452
  38. 3
  39. 6281
  40. 1

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