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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 573 · 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 447 to make it divisible by 9? (a) 3 (b) 6 (c) 2 (d) 4
  2. 2.What is the remainder when 588 is divided by 10? (a) 9 (b) 2 (c) 7 (d) 8
  3. 3.What is the remainder when 484 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
  4. 4.What is the units digit of 3^27? (a) 9 (b) 1 (c) 3 (d) 7
  5. 5.What is the smallest number you must add to 333 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  6. 6.What is the remainder when 638 is divided by 12? (a) 1 (b) 2 (c) 3 (d) 10
  7. 7.What is the smallest number you must add to 61 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  8. 8.What is the remainder when 97 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
  9. 9.What is the smallest number you must add to 382 to make it divisible by 10? (a) 2 (b) 9 (c) 8 (d) 7
  10. 10.Which of these numbers is divisible by 9? (a) 7281 (b) 7277 (c) 7285 (d) 7298
  11. 11.What is the smallest number you must add to 195 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
  12. 12.What is the remainder when 549 is divided by 12? (a) 9 (b) 8 (c) 3 (d) 10
  13. 13.What is the smallest number you must add to 151 to make it divisible by 3? (a) 2 (b) 1
  14. 14.Which of these numbers is divisible by 9? (a) 694 (b) 684 (c) 677 (d) 692
  15. 15.What is the smallest number you must add to 196 to make it divisible by 3? (a) 1 (b) 2
  16. 16.Which of these numbers is divisible by 8? (a) 2243 (b) 2250 (c) 2258 (d) 2248
  17. 17.What is the units digit of 3^16? (a) 3 (b) 7 (c) 9 (d) 1
  18. 18.What is the smallest number you must add to 35 to make it divisible by 3? (a) 2 (b) 1
  19. 19.Which of these numbers is divisible by 8? (a) 4379 (b) 4375 (c) 4376 (d) 4385
  20. 20.Which of these numbers is divisible by 6? (a) 2007 (b) 2011 (c) 2004 (d) 1999
  21. 21.What is the smallest number you must add to 206 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
  22. 22.What is the smallest number you must add to 436 to make it divisible by 7? (a) 5 (b) 6 (c) 2 (d) 4
  23. 23.What is the smallest number you must add to 175 to make it divisible by 12? (a) 4 (b) 7 (c) 6 (d) 5
  24. 24.Which of these numbers is divisible by 3? (a) 2227 (b) 2221 (c) 2223 (d) 2225
  25. 25.What is the remainder when 227 is divided by 9? (a) 3 (b) 2 (c) 7 (d) 1
  26. 26.What is the remainder when 191 is divided by 6? (a) 5 (b) 1 (c) 3 (d) 4
  27. 27.What is the units digit of 7^58? (a) 7 (b) 9 (c) 3 (d) 1
  28. 28.What is the remainder when 185 is divided by 9? (a) 7 (b) 4 (c) 5 (d) 6
  29. 29.Which of these numbers is divisible by 8? (a) 4969 (b) 4960 (c) 4961 (d) 4958
  30. 30.What is the remainder when 122 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
  31. 31.What is the remainder when 103 is divided by 6? (a) 0 (b) 5 (c) 2 (d) 1
  32. 32.What is the units digit of 9^48? (a) 9 (b) 0 (c) 2 (d) 1
  33. 33.Which of these numbers is divisible by 4? (a) 2337 (b) 2330 (c) 2333 (d) 2332
  34. 34.What is the smallest number you must add to 272 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  35. 35.What is the remainder when 345 is divided by 9? (a) 6 (b) 4 (c) 3 (d) 2
  36. 36.What is the remainder when 393 is divided by 10? (a) 3 (b) 2 (c) 7 (d) 4
  37. 37.Which of these numbers is divisible by 4? (a) 2508 (b) 2513 (c) 2506 (d) 2509
  38. 38.What is the remainder when 545 is divided by 12? (a) 7 (b) 5 (c) 6 (d) 4
  39. 39.What is the remainder when 543 is divided by 8? (a) 1 (b) 6 (c) 7 (d) 5
  40. 40.What is the units digit of 8^56? (a) 8 (b) 2 (c) 4 (d) 6

Answer key

  1. 3
  2. 8
  3. 1
  4. 7
  5. 2
  6. 2
  7. 4
  8. 1
  9. 8
  10. 7281
  11. 5
  12. 9
  13. 2
  14. 684
  15. 2
  16. 2248
  17. 1
  18. 1
  19. 4376
  20. 2004
  21. 4
  22. 5
  23. 5
  24. 2223
  25. 2
  26. 5
  27. 9
  28. 5
  29. 4960
  30. 2
  31. 1
  32. 1
  33. 2332
  34. 3
  35. 3
  36. 3
  37. 2508
  38. 5
  39. 7
  40. 6

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