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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 98 · 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 9^45? (a) 1 (b) 7 (c) 8 (d) 9
  2. 2.What is the units digit of 9^20? (a) 1 (b) 2 (c) 9 (d) 0
  3. 3.Which of these numbers is divisible by 4? (a) 2525 (b) 2530 (c) 2528 (d) 2533
  4. 4.What is the smallest number you must add to 341 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 1
  5. 5.What is the smallest number you must add to 160 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
  6. 6.What is the remainder when 120 is divided by 7? (a) 0 (b) 6 (c) 2 (d) 1
  7. 7.What is the smallest number you must add to 271 to make it divisible by 8? (a) 7 (b) 3 (c) 2 (d) 1
  8. 8.What is the smallest number you must add to 682 to make it divisible by 10? (a) 9 (b) 2 (c) 7 (d) 8
  9. 9.What is the remainder when 224 is divided by 9? (a) 6 (b) 7 (c) 1 (d) 8
  10. 10.What is the units digit of 2^55? (a) 4 (b) 2 (c) 6 (d) 8
  11. 11.Which of these numbers is divisible by 6? (a) 258 (b) 259 (c) 269 (d) 253
  12. 12.What is the units digit of 7^49? (a) 1 (b) 3 (c) 9 (d) 7
  13. 13.What is the remainder when 327 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
  14. 14.Which of these numbers is divisible by 8? (a) 1806 (b) 1788 (c) 1792 (d) 1794
  15. 15.What is the remainder when 141 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 7
  16. 16.What is the remainder when 71 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
  17. 17.What is the smallest number you must add to 732 to make it divisible by 10? (a) 9 (b) 2 (c) 7 (d) 8
  18. 18.Which of these numbers is divisible by 3? (a) 626 (b) 632 (c) 627 (d) 629
  19. 19.What is the smallest number you must add to 304 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  20. 20.What is the smallest number you must add to 233 to make it divisible by 9? (a) 8 (b) 2 (c) 1 (d) 3
  21. 21.What is the remainder when 353 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
  22. 22.Which of these numbers is divisible by 4? (a) 3441 (b) 3449 (c) 3446 (d) 3444
  23. 23.Which of these numbers is divisible by 3? (a) 2017 (b) 2021 (c) 2014 (d) 2016
  24. 24.Which of these numbers is divisible by 3? (a) 1646 (b) 1639 (c) 1643 (d) 1641
  25. 25.Which of these numbers is divisible by 11? (a) 2967 (b) 2948 (c) 2943 (d) 2949
  26. 26.What is the remainder when 293 is divided by 12? (a) 5 (b) 7 (c) 4 (d) 6
  27. 27.What is the remainder when 199 is divided by 11? (a) 2 (b) 0 (c) 10 (d) 1
  28. 28.Which of these numbers is divisible by 9? (a) 5283 (b) 5280 (c) 5300 (d) 5286
  29. 29.What is the units digit of 2^27? (a) 6 (b) 4 (c) 8 (d) 2
  30. 30.Which of these numbers is divisible by 3? (a) 709 (b) 708 (c) 707 (d) 712
  31. 31.What is the remainder when 101 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
  32. 32.What is the smallest number you must add to 241 to make it divisible by 3? (a) 2 (b) 1
  33. 33.Which of these numbers is divisible by 6? (a) 3822 (b) 3833 (c) 3824 (d) 3821
  34. 34.What is the remainder when 558 is divided by 7? (a) 2 (b) 5 (c) 6 (d) 4
  35. 35.What is the smallest number you must add to 234 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
  36. 36.What is the remainder when 158 is divided by 8? (a) 6 (b) 5 (c) 7 (d) 2
  37. 37.Which of these numbers is divisible by 11? (a) 1453 (b) 1463 (c) 1475 (d) 1472
  38. 38.What is the remainder when 368 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 3
  39. 39.Which of these numbers is divisible by 8? (a) 5382 (b) 5374 (c) 5364 (d) 5368
  40. 40.What is the smallest number you must add to 371 to make it divisible by 9? (a) 6 (b) 2 (c) 7 (d) 8

Answer key

  1. 9
  2. 1
  3. 2528
  4. 1
  5. 2
  6. 1
  7. 1
  8. 8
  9. 8
  10. 8
  11. 258
  12. 7
  13. 2
  14. 1792
  15. 6
  16. 1
  17. 8
  18. 627
  19. 1
  20. 1
  21. 3
  22. 3444
  23. 2016
  24. 1641
  25. 2948
  26. 5
  27. 1
  28. 5283
  29. 8
  30. 708
  31. 1
  32. 2
  33. 3822
  34. 5
  35. 4
  36. 6
  37. 1463
  38. 4
  39. 5368
  40. 7

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