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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 565 · 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 166 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  2. 2.What is the smallest number you must add to 153 to make it divisible by 8? (a) 1 (b) 6 (c) 5 (d) 7
  3. 3.Which of these numbers is divisible by 8? (a) 5417 (b) 5424 (c) 5425 (d) 5438
  4. 4.What is the units digit of 4^52? (a) 4 (b) 7 (c) 5 (d) 6
  5. 5.What is the units digit of 3^18? (a) 3 (b) 9 (c) 1 (d) 7
  6. 6.Which of these numbers is divisible by 8? (a) 1889 (b) 1900 (c) 1888 (d) 1884
  7. 7.What is the smallest number you must add to 169 to make it divisible by 3? (a) 2 (b) 1
  8. 8.What is the smallest number you must add to 244 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
  9. 9.What is the remainder when 504 is divided by 10? (a) 3 (b) 5 (c) 4 (d) 6
  10. 10.What is the remainder when 125 is divided by 3? (a) 0 (b) 2 (c) 1
  11. 11.What is the smallest number you must add to 497 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 2
  12. 12.What is the smallest number you must add to 616 to make it divisible by 12? (a) 7 (b) 9 (c) 4 (d) 8
  13. 13.What is the smallest number you must add to 136 to make it divisible by 3? (a) 1 (b) 2
  14. 14.What is the remainder when 282 is divided by 7? (a) 3 (b) 1 (c) 5 (d) 2
  15. 15.What is the smallest number you must add to 651 to make it divisible by 9? (a) 6 (b) 7 (c) 3 (d) 5
  16. 16.What is the smallest number you must add to 49 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  17. 17.Which of these numbers is divisible by 6? (a) 3296 (b) 3290 (c) 3305 (d) 3294
  18. 18.What is the smallest number you must add to 259 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  19. 19.What is the remainder when 87 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
  20. 20.What is the units digit of 7^16? (a) 3 (b) 1 (c) 7 (d) 9
  21. 21.What is the smallest number you must add to 119 to make it divisible by 3? (a) 1 (b) 2
  22. 22.Which of these numbers is divisible by 8? (a) 6803 (b) 6815 (c) 6800 (d) 6794
  23. 23.Which of these numbers is divisible by 9? (a) 443 (b) 433 (c) 441 (d) 454
  24. 24.Which of these numbers is divisible by 4? (a) 3163 (b) 3171 (c) 3167 (d) 3164
  25. 25.Which of these numbers is divisible by 3? (a) 160 (b) 157 (c) 164 (d) 159
  26. 26.What is the smallest number you must add to 182 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
  27. 27.Which of these numbers is divisible by 3? (a) 1074 (b) 1072 (c) 1078 (d) 1075
  28. 28.What is the units digit of 8^54? (a) 8 (b) 6 (c) 2 (d) 4
  29. 29.What is the smallest number you must add to 437 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
  30. 30.What is the smallest number you must add to 512 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 2
  31. 31.What is the remainder when 164 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
  32. 32.What is the smallest number you must add to 196 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  33. 33.Which of these numbers is divisible by 4? (a) 1074 (b) 1065 (c) 1068 (d) 1069
  34. 34.What is the remainder when 113 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
  35. 35.What is the remainder when 341 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
  36. 36.What is the units digit of 4^43? (a) 6 (b) 3 (c) 5 (d) 4
  37. 37.What is the remainder when 345 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
  38. 38.Which of these numbers is divisible by 8? (a) 2312 (b) 2311 (c) 2322 (d) 2319
  39. 39.What is the units digit of 4^35? (a) 6 (b) 4 (c) 3 (d) 5
  40. 40.Which of these numbers is divisible by 9? (a) 4107 (b) 4089 (c) 4103 (d) 4095

Answer key

  1. 4
  2. 7
  3. 5424
  4. 6
  5. 9
  6. 1888
  7. 2
  8. 8
  9. 4
  10. 2
  11. 7
  12. 8
  13. 2
  14. 2
  15. 6
  16. 3
  17. 3294
  18. 1
  19. 3
  20. 1
  21. 1
  22. 6800
  23. 441
  24. 3164
  25. 159
  26. 4
  27. 1074
  28. 4
  29. 4
  30. 8
  31. 4
  32. 4
  33. 1068
  34. 5
  35. 5
  36. 4
  37. 3
  38. 2312
  39. 4
  40. 4095

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