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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 352 · 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 668 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 2
  2. 2.Which of these numbers is divisible by 9? (a) 7260 (b) 7263 (c) 7264 (d) 7275
  3. 3.Which of these numbers is divisible by 3? (a) 1333 (b) 1328 (c) 1329 (d) 1330
  4. 4.What is the remainder when 441 is divided by 8? (a) 7 (b) 1 (c) 2 (d) 0
  5. 5.What is the smallest number you must add to 215 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  6. 6.Which of these numbers is divisible by 11? (a) 3223 (b) 3222 (c) 3238 (d) 3232
  7. 7.What is the smallest number you must add to 234 to make it divisible by 7? (a) 6 (b) 3 (c) 5 (d) 4
  8. 8.What is the smallest number you must add to 884 to make it divisible by 12? (a) 5 (b) 3 (c) 8 (d) 4
  9. 9.What is the remainder when 139 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 4
  10. 10.Which of these numbers is divisible by 4? (a) 1922 (b) 1914 (c) 1918 (d) 1916
  11. 11.What is the units digit of 4^52? (a) 5 (b) 6 (c) 4 (d) 7
  12. 12.What is the smallest number you must add to 501 to make it divisible by 8? (a) 2 (b) 5 (c) 4 (d) 3
  13. 13.What is the smallest number you must add to 293 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  14. 14.What is the smallest number you must add to 190 to make it divisible by 7? (a) 6 (b) 1 (c) 5 (d) 4
  15. 15.What is the units digit of 3^15? (a) 9 (b) 7 (c) 3 (d) 1
  16. 16.What is the remainder when 243 is divided by 11? (a) 10 (b) 2 (c) 0 (d) 1
  17. 17.Which of these numbers is divisible by 3? (a) 659 (b) 662 (c) 665 (d) 660
  18. 18.What is the smallest number you must add to 700 to make it divisible by 12? (a) 4 (b) 9 (c) 7 (d) 8
  19. 19.What is the smallest number you must add to 175 to make it divisible by 3? (a) 1 (b) 2
  20. 20.What is the smallest number you must add to 457 to make it divisible by 12? (a) 10 (b) 9 (c) 11 (d) 1
  21. 21.What is the remainder when 515 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
  22. 22.What is the smallest number you must add to 749 to make it divisible by 11? (a) 9 (b) 8 (c) 10 (d) 1
  23. 23.What is the units digit of 3^17? (a) 3 (b) 7 (c) 9 (d) 1
  24. 24.What is the units digit of 9^23? (a) 8 (b) 7 (c) 9 (d) 1
  25. 25.Which of these numbers is divisible by 9? (a) 1662 (b) 1680 (c) 1672 (d) 1665
  26. 26.What is the remainder when 159 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 6
  27. 27.Which of these numbers is divisible by 6? (a) 2354 (b) 2351 (c) 2359 (d) 2352
  28. 28.What is the remainder when 288 is divided by 10? (a) 7 (b) 9 (c) 8 (d) 2
  29. 29.What is the smallest number you must add to 776 to make it divisible by 12? (a) 8 (b) 3 (c) 4 (d) 5
  30. 30.Which of these numbers is divisible by 6? (a) 2608 (b) 2621 (c) 2610 (d) 2615
  31. 31.What is the smallest number you must add to 344 to make it divisible by 12? (a) 3 (b) 5 (c) 8 (d) 4
  32. 32.What is the remainder when 124 is divided by 3? (a) 0 (b) 2 (c) 1
  33. 33.What is the remainder when 101 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
  34. 34.What is the remainder when 133 is divided by 3? (a) 1 (b) 2 (c) 0
  35. 35.What is the smallest number you must add to 675 to make it divisible by 10? (a) 7 (b) 4 (c) 6 (d) 5
  36. 36.What is the units digit of 4^18? (a) 7 (b) 5 (c) 4 (d) 6
  37. 37.Which of these numbers is divisible by 4? (a) 1957 (b) 1962 (c) 1960 (d) 1965
  38. 38.What is the units digit of 2^43? (a) 6 (b) 8 (c) 2 (d) 4
  39. 39.What is the units digit of 2^48? (a) 4 (b) 8 (c) 2 (d) 6
  40. 40.Which of these numbers is divisible by 11? (a) 3254 (b) 3234 (c) 3232 (d) 3244

Answer key

  1. 7
  2. 7263
  3. 1329
  4. 1
  5. 1
  6. 3223
  7. 4
  8. 4
  9. 4
  10. 1916
  11. 6
  12. 3
  13. 3
  14. 6
  15. 7
  16. 1
  17. 660
  18. 8
  19. 2
  20. 11
  21. 5
  22. 10
  23. 3
  24. 9
  25. 1665
  26. 5
  27. 2352
  28. 8
  29. 4
  30. 2610
  31. 4
  32. 1
  33. 1
  34. 1
  35. 5
  36. 6
  37. 1960
  38. 8
  39. 6
  40. 3234

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