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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 339 · 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 remainder when 74 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
  2. 2.What is the remainder when 562 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
  3. 3.What is the remainder when 111 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
  4. 4.What is the remainder when 255 is divided by 7? (a) 3 (b) 2 (c) 5 (d) 4
  5. 5.Which of these numbers is divisible by 3? (a) 1168 (b) 1167 (c) 1171 (d) 1166
  6. 6.What is the units digit of 4^57? (a) 6 (b) 5 (c) 3 (d) 4
  7. 7.What is the units digit of 8^22? (a) 4 (b) 2 (c) 6 (d) 8
  8. 8.What is the smallest number you must add to 321 to make it divisible by 7? (a) 3 (b) 6 (c) 1 (d) 2
  9. 9.What is the smallest number you must add to 129 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
  10. 10.Which of these numbers is divisible by 4? (a) 1553 (b) 1548 (c) 1546 (d) 1549
  11. 11.What is the units digit of 2^44? (a) 2 (b) 8 (c) 4 (d) 6
  12. 12.What is the smallest number you must add to 189 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
  13. 13.Which of these numbers is divisible by 3? (a) 2330 (b) 2327 (c) 2332 (d) 2328
  14. 14.Which of these numbers is divisible by 9? (a) 6919 (b) 6912 (c) 6909 (d) 6925
  15. 15.What is the units digit of 9^20? (a) 1 (b) 9 (c) 0 (d) 2
  16. 16.What is the smallest number you must add to 491 to make it divisible by 9? (a) 3 (b) 4 (c) 5 (d) 6
  17. 17.What is the remainder when 531 is divided by 8? (a) 5 (b) 4 (c) 2 (d) 3
  18. 18.What is the smallest number you must add to 129 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  19. 19.What is the remainder when 218 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
  20. 20.What is the remainder when 736 is divided by 12? (a) 3 (b) 8 (c) 4 (d) 5
  21. 21.What is the units digit of 2^24? (a) 8 (b) 6 (c) 2 (d) 4
  22. 22.What is the units digit of 4^18? (a) 6 (b) 4 (c) 5 (d) 7
  23. 23.What is the units digit of 9^40? (a) 9 (b) 2 (c) 0 (d) 1
  24. 24.What is the smallest number you must add to 178 to make it divisible by 9? (a) 3 (b) 7 (c) 1 (d) 2
  25. 25.What is the remainder when 156 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
  26. 26.What is the smallest number you must add to 201 to make it divisible by 7? (a) 1 (b) 3 (c) 5 (d) 2
  27. 27.What is the smallest number you must add to 128 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2
  28. 28.Which of these numbers is divisible by 3? (a) 1531 (b) 1529 (c) 1527 (d) 1526
  29. 29.Which of these numbers is divisible by 3? (a) 367 (b) 366 (c) 364 (d) 371
  30. 30.Which of these numbers is divisible by 6? (a) 1317 (b) 1314 (c) 1310 (d) 1321
  31. 31.What is the remainder when 538 is divided by 9? (a) 2 (b) 8 (c) 7 (d) 6
  32. 32.Which of these numbers is divisible by 4? (a) 2465 (b) 2461 (c) 2469 (d) 2464
  33. 33.What is the smallest number you must add to 70 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 2
  34. 34.What is the remainder when 259 is divided by 6? (a) 0 (b) 1 (c) 2 (d) 5
  35. 35.What is the smallest number you must add to 644 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
  36. 36.What is the smallest number you must add to 323 to make it divisible by 10? (a) 8 (b) 6 (c) 7 (d) 3
  37. 37.What is the remainder when 207 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  38. 38.What is the smallest number you must add to 93 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  39. 39.What is the smallest number you must add to 295 to make it divisible by 9? (a) 2 (b) 7 (c) 1 (d) 3
  40. 40.Which of these numbers is divisible by 3? (a) 1016 (b) 1013 (c) 1014 (d) 1018

Answer key

  1. 4
  2. 2
  3. 1
  4. 3
  5. 1167
  6. 4
  7. 4
  8. 1
  9. 3
  10. 1548
  11. 6
  12. 3
  13. 2328
  14. 6912
  15. 1
  16. 4
  17. 3
  18. 3
  19. 2
  20. 4
  21. 6
  22. 6
  23. 1
  24. 2
  25. 2
  26. 2
  27. 5
  28. 1527
  29. 366
  30. 1314
  31. 7
  32. 2464
  33. 2
  34. 1
  35. 4
  36. 7
  37. 2
  38. 3
  39. 2
  40. 1014

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