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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 191 · 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.Which of these numbers is divisible by 3? (a) 580 (b) 582 (c) 587 (d) 584
  2. 2.What is the smallest number you must add to 179 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  3. 3.Which of these numbers is divisible by 9? (a) 4611 (b) 4617 (c) 4622 (d) 4632
  4. 4.What is the smallest number you must add to 139 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  5. 5.Which of these numbers is divisible by 4? (a) 2037 (b) 2035 (c) 2041 (d) 2036
  6. 6.Which of these numbers is divisible by 3? (a) 519 (b) 517 (c) 521 (d) 523
  7. 7.Which of these numbers is divisible by 3? (a) 1507 (b) 1501 (c) 1504 (d) 1503
  8. 8.What is the smallest number you must add to 269 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 5
  9. 9.What is the remainder when 102 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 6
  10. 10.Which of these numbers is divisible by 9? (a) 3890 (b) 3877 (c) 3879 (d) 3880
  11. 11.What is the smallest number you must add to 126 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  12. 12.What is the smallest number you must add to 194 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  13. 13.What is the smallest number you must add to 439 to make it divisible by 7? (a) 1 (b) 3 (c) 5 (d) 2
  14. 14.What is the units digit of 3^50? (a) 1 (b) 3 (c) 9 (d) 7
  15. 15.Which of these numbers is divisible by 3? (a) 1313 (b) 1306 (c) 1308 (d) 1309
  16. 16.What is the units digit of 4^60? (a) 4 (b) 7 (c) 5 (d) 6
  17. 17.What is the remainder when 73 is divided by 3? (a) 1 (b) 2 (c) 0
  18. 18.What is the smallest number you must add to 435 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
  19. 19.Which of these numbers is divisible by 3? (a) 2026 (b) 2029 (c) 2032 (d) 2028
  20. 20.Which of these numbers is divisible by 11? (a) 6888 (b) 6917 (c) 6897 (d) 6901
  21. 21.What is the units digit of 2^12? (a) 4 (b) 2 (c) 6 (d) 8
  22. 22.Which of these numbers is divisible by 3? (a) 815 (b) 817 (c) 816 (d) 820
  23. 23.What is the smallest number you must add to 404 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  24. 24.What is the smallest number you must add to 634 to make it divisible by 12? (a) 3 (b) 2 (c) 10 (d) 1
  25. 25.What is the units digit of 2^58? (a) 8 (b) 4 (c) 6 (d) 2
  26. 26.Which of these numbers is divisible by 3? (a) 799 (b) 798 (c) 802 (d) 796
  27. 27.What is the smallest number you must add to 352 to make it divisible by 9? (a) 7 (b) 1 (c) 8 (d) 6
  28. 28.What is the smallest number you must add to 242 to make it divisible by 3? (a) 2 (b) 1
  29. 29.Which of these numbers is divisible by 11? (a) 7357 (b) 7377 (c) 7359 (d) 7369
  30. 30.What is the units digit of 7^11? (a) 3 (b) 7 (c) 1 (d) 9
  31. 31.What is the smallest number you must add to 314 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  32. 32.What is the units digit of 3^36? (a) 9 (b) 1 (c) 7 (d) 3
  33. 33.What is the units digit of 4^54? (a) 5 (b) 4 (c) 7 (d) 6
  34. 34.Which of these numbers is divisible by 9? (a) 1737 (b) 1742 (c) 1735 (d) 1751
  35. 35.What is the units digit of 4^30? (a) 7 (b) 6 (c) 4 (d) 5
  36. 36.What is the smallest number you must add to 131 to make it divisible by 3? (a) 1 (b) 2
  37. 37.What is the smallest number you must add to 422 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
  38. 38.What is the units digit of 3^52? (a) 3 (b) 7 (c) 9 (d) 1
  39. 39.Which of these numbers is divisible by 4? (a) 297 (b) 296 (c) 301 (d) 295
  40. 40.What is the smallest number you must add to 274 to make it divisible by 4? (a) 3 (b) 2 (c) 1

Answer key

  1. 582
  2. 1
  3. 4617
  4. 1
  5. 2036
  6. 519
  7. 1503
  8. 1
  9. 4
  10. 3879
  11. 2
  12. 2
  13. 2
  14. 9
  15. 1308
  16. 6
  17. 1
  18. 3
  19. 2028
  20. 6897
  21. 6
  22. 816
  23. 1
  24. 2
  25. 4
  26. 798
  27. 8
  28. 1
  29. 7359
  30. 3
  31. 1
  32. 1
  33. 6
  34. 1737
  35. 6
  36. 1
  37. 4
  38. 1
  39. 296
  40. 2

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