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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 513 · 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 11? (a) 1485 (b) 1498 (c) 1483 (d) 1492
  2. 2.What is the smallest number you must add to 295 to make it divisible by 9? (a) 7 (b) 1 (c) 2 (d) 3
  3. 3.What is the smallest number you must add to 128 to make it divisible by 3? (a) 1 (b) 2
  4. 4.What is the smallest number you must add to 101 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 2
  5. 5.What is the remainder when 85 is divided by 3? (a) 1 (b) 0 (c) 2
  6. 6.What is the remainder when 155 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  7. 7.What is the remainder when 353 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 3
  8. 8.What is the remainder when 401 is divided by 12? (a) 4 (b) 5 (c) 7 (d) 6
  9. 9.Which of these numbers is divisible by 9? (a) 7346 (b) 7336 (c) 7361 (d) 7344
  10. 10.What is the smallest number you must add to 435 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
  11. 11.What is the units digit of 4^20? (a) 7 (b) 5 (c) 6 (d) 4
  12. 12.Which of these numbers is divisible by 9? (a) 666 (b) 682 (c) 662 (d) 672
  13. 13.What is the smallest number you must add to 285 to make it divisible by 11? (a) 1 (b) 3 (c) 10 (d) 2
  14. 14.What is the smallest number you must add to 256 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
  15. 15.Which of these numbers is divisible by 3? (a) 1301 (b) 1307 (c) 1302 (d) 1303
  16. 16.What is the smallest number you must add to 87 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  17. 17.Which of these numbers is divisible by 6? (a) 564 (b) 565 (c) 563 (d) 575
  18. 18.Which of these numbers is divisible by 9? (a) 4611 (b) 4599 (c) 4593 (d) 4600
  19. 19.What is the units digit of 2^13? (a) 6 (b) 2 (c) 8 (d) 4
  20. 20.Which of these numbers is divisible by 4? (a) 3038 (b) 3046 (c) 3040 (d) 3042
  21. 21.What is the remainder when 294 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
  22. 22.What is the smallest number you must add to 258 to make it divisible by 9? (a) 3 (b) 2 (c) 6 (d) 4
  23. 23.What is the remainder when 716 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
  24. 24.What is the remainder when 605 is divided by 12? (a) 6 (b) 5 (c) 4 (d) 7
  25. 25.Which of these numbers is divisible by 4? (a) 1234 (b) 1236 (c) 1241 (d) 1239
  26. 26.What is the smallest number you must add to 673 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
  27. 27.Which of these numbers is divisible by 6? (a) 386 (b) 379 (c) 394 (d) 384
  28. 28.What is the remainder when 84 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  29. 29.What is the remainder when 782 is divided by 11? (a) 10 (b) 2 (c) 0 (d) 1
  30. 30.What is the units digit of 3^14? (a) 9 (b) 7 (c) 1 (d) 3
  31. 31.What is the remainder when 86 is divided by 3? (a) 0 (b) 2 (c) 1
  32. 32.Which of these numbers is divisible by 8? (a) 3500 (b) 3496 (c) 3511 (d) 3495
  33. 33.What is the remainder when 363 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
  34. 34.What is the remainder when 614 is divided by 9? (a) 7 (b) 3 (c) 1 (d) 2
  35. 35.What is the smallest number you must add to 177 to make it divisible by 12? (a) 4 (b) 2 (c) 3 (d) 9
  36. 36.What is the units digit of 4^53? (a) 4 (b) 3 (c) 6 (d) 5
  37. 37.What is the smallest number you must add to 152 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 9
  38. 38.What is the remainder when 229 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
  39. 39.What is the remainder when 118 is divided by 3? (a) 0 (b) 2 (c) 1
  40. 40.What is the smallest number you must add to 100 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5

Answer key

  1. 1485
  2. 2
  3. 1
  4. 1
  5. 1
  6. 3
  7. 3
  8. 5
  9. 7344
  10. 3
  11. 6
  12. 666
  13. 1
  14. 2
  15. 1302
  16. 3
  17. 564
  18. 4599
  19. 2
  20. 3040
  21. 2
  22. 3
  23. 6
  24. 5
  25. 1236
  26. 9
  27. 384
  28. 4
  29. 1
  30. 9
  31. 2
  32. 3496
  33. 3
  34. 2
  35. 3
  36. 4
  37. 2
  38. 4
  39. 1
  40. 4

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