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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 809 · 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 units digit of 3^54? (a) 3 (b) 9 (c) 7 (d) 1
  2. 2.What is the smallest number you must add to 721 to make it divisible by 12? (a) 10 (b) 1 (c) 11 (d) 9
  3. 3.What is the smallest number you must add to 234 to make it divisible by 7? (a) 3 (b) 6 (c) 5 (d) 4
  4. 4.Which of these numbers is divisible by 9? (a) 4525 (b) 4516 (c) 4508 (d) 4509
  5. 5.Which of these numbers is divisible by 3? (a) 461 (b) 464 (c) 457 (d) 459
  6. 6.What is the units digit of 7^60? (a) 1 (b) 7 (c) 3 (d) 9
  7. 7.What is the smallest number you must add to 425 to make it divisible by 9? (a) 2 (b) 8 (c) 6 (d) 7
  8. 8.Which of these numbers is divisible by 11? (a) 5376 (b) 5380 (c) 5379 (d) 5398
  9. 9.Which of these numbers is divisible by 3? (a) 2270 (b) 2268 (c) 2272 (d) 2267
  10. 10.Which of these numbers is divisible by 8? (a) 424 (b) 428 (c) 433 (d) 422
  11. 11.What is the smallest number you must add to 233 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  12. 12.What is the remainder when 205 is divided by 6? (a) 2 (b) 1 (c) 5 (d) 0
  13. 13.Which of these numbers is divisible by 11? (a) 3043 (b) 3036 (c) 3034 (d) 3051
  14. 14.What is the units digit of 8^41? (a) 2 (b) 6 (c) 8 (d) 4
  15. 15.What is the smallest number you must add to 354 to make it divisible by 9? (a) 7 (b) 3 (c) 6 (d) 5
  16. 16.What is the units digit of 2^48? (a) 4 (b) 2 (c) 8 (d) 6
  17. 17.What is the smallest number you must add to 323 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
  18. 18.What is the smallest number you must add to 254 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
  19. 19.What is the smallest number you must add to 386 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 2
  20. 20.Which of these numbers is divisible by 3? (a) 1165 (b) 1169 (c) 1171 (d) 1167
  21. 21.Which of these numbers is divisible by 4? (a) 2238 (b) 2232 (c) 2230 (d) 2233
  22. 22.What is the remainder when 462 is divided by 8? (a) 2 (b) 5 (c) 7 (d) 6
  23. 23.Which of these numbers is divisible by 8? (a) 6402 (b) 6396 (c) 6392 (d) 6389
  24. 24.What is the smallest number you must add to 242 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 2
  25. 25.What is the smallest number you must add to 389 to make it divisible by 12? (a) 8 (b) 7 (c) 6 (d) 5
  26. 26.Which of these numbers is divisible by 9? (a) 7774 (b) 7776 (c) 7779 (d) 7789
  27. 27.Which of these numbers is divisible by 11? (a) 6779 (b) 6789 (c) 6776 (d) 6769
  28. 28.Which of these numbers is divisible by 3? (a) 1763 (b) 1764 (c) 1769 (d) 1766
  29. 29.What is the units digit of 7^59? (a) 9 (b) 7 (c) 1 (d) 3
  30. 30.Which of these numbers is divisible by 3? (a) 1582 (b) 1589 (c) 1584 (d) 1585
  31. 31.Which of these numbers is divisible by 9? (a) 6574 (b) 6564 (c) 6584 (d) 6570
  32. 32.What is the remainder when 535 is divided by 11? (a) 7 (b) 6 (c) 8 (d) 4
  33. 33.What is the remainder when 228 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
  34. 34.What is the remainder when 90 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
  35. 35.What is the smallest number you must add to 309 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 9
  36. 36.Which of these numbers is divisible by 8? (a) 1104 (b) 1099 (c) 1116 (d) 1106
  37. 37.Which of these numbers is divisible by 6? (a) 3537 (b) 3548 (c) 3545 (d) 3540
  38. 38.What is the smallest number you must add to 287 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
  39. 39.Which of these numbers is divisible by 11? (a) 847 (b) 837 (c) 865 (d) 851
  40. 40.Which of these numbers is divisible by 9? (a) 3825 (b) 3838 (c) 3822 (d) 3833

Answer key

  1. 9
  2. 11
  3. 4
  4. 4509
  5. 459
  6. 1
  7. 7
  8. 5379
  9. 2268
  10. 424
  11. 3
  12. 1
  13. 3036
  14. 8
  15. 6
  16. 6
  17. 6
  18. 4
  19. 6
  20. 1167
  21. 2232
  22. 6
  23. 6392
  24. 10
  25. 7
  26. 7776
  27. 6776
  28. 1764
  29. 3
  30. 1584
  31. 6570
  32. 7
  33. 3
  34. 2
  35. 1
  36. 1104
  37. 3540
  38. 3
  39. 847
  40. 3825

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