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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 608 · 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 149 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  2. 2.What is the units digit of 7^33? (a) 9 (b) 1 (c) 7 (d) 3
  3. 3.What is the units digit of 8^31? (a) 4 (b) 6 (c) 8 (d) 2
  4. 4.What is the remainder when 153 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
  5. 5.What is the remainder when 221 is divided by 3? (a) 0 (b) 2 (c) 1
  6. 6.What is the remainder when 341 is divided by 8? (a) 3 (b) 4 (c) 5 (d) 6
  7. 7.What is the remainder when 234 is divided by 11? (a) 8 (b) 2 (c) 4 (d) 3
  8. 8.What is the smallest number you must add to 358 to make it divisible by 8? (a) 1 (b) 6 (c) 3 (d) 2
  9. 9.Which of these numbers is divisible by 3? (a) 1742 (b) 1743 (c) 1748 (d) 1744
  10. 10.What is the smallest number you must add to 84 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4
  11. 11.Which of these numbers is divisible by 6? (a) 4349 (b) 4341 (c) 4333 (d) 4338
  12. 12.What is the smallest number you must add to 543 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 7
  13. 13.What is the smallest number you must add to 526 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
  14. 14.Which of these numbers is divisible by 9? (a) 6418 (b) 6412 (c) 6417 (d) 6428
  15. 15.What is the remainder when 705 is divided by 12? (a) 9 (b) 8 (c) 10 (d) 3
  16. 16.What is the remainder when 354 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 6
  17. 17.What is the remainder when 484 is divided by 9? (a) 7 (b) 2 (c) 6 (d) 8
  18. 18.Which of these numbers is divisible by 6? (a) 3404 (b) 3409 (c) 3399 (d) 3402
  19. 19.What is the smallest number you must add to 315 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
  20. 20.What is the remainder when 43 is divided by 3? (a) 2 (b) 0 (c) 1
  21. 21.What is the units digit of 2^49? (a) 4 (b) 8 (c) 6 (d) 2
  22. 22.What is the smallest number you must add to 573 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3
  23. 23.What is the smallest number you must add to 139 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  24. 24.What is the units digit of 9^52? (a) 1 (b) 0 (c) 9 (d) 2
  25. 25.What is the remainder when 82 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 6
  26. 26.Which of these numbers is divisible by 4? (a) 262 (b) 255 (c) 256 (d) 258
  27. 27.What is the remainder when 647 is divided by 8? (a) 5 (b) 7 (c) 1 (d) 6
  28. 28.Which of these numbers is divisible by 8? (a) 758 (b) 746 (c) 744 (d) 740
  29. 29.What is the units digit of 4^11? (a) 6 (b) 3 (c) 5 (d) 4
  30. 30.What is the remainder when 428 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 1
  31. 31.Which of these numbers is divisible by 4? (a) 1095 (b) 1085 (c) 1091 (d) 1088
  32. 32.What is the units digit of 3^51? (a) 7 (b) 3 (c) 1 (d) 9
  33. 33.What is the smallest number you must add to 257 to make it divisible by 6? (a) 5 (b) 1 (c) 2 (d) 3
  34. 34.Which of these numbers is divisible by 3? (a) 277 (b) 271 (c) 274 (d) 273
  35. 35.Which of these numbers is divisible by 6? (a) 4838 (b) 4850 (c) 4843 (d) 4842
  36. 36.What is the remainder when 147 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
  37. 37.Which of these numbers is divisible by 3? (a) 446 (b) 449 (c) 444 (d) 442
  38. 38.Which of these numbers is divisible by 4? (a) 456 (b) 457 (c) 462 (d) 454
  39. 39.What is the smallest number you must add to 114 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  40. 40.What is the smallest number you must add to 135 to make it divisible by 4? (a) 3 (b) 2 (c) 1

Answer key

  1. 1
  2. 7
  3. 2
  4. 1
  5. 2
  6. 5
  7. 3
  8. 2
  9. 1743
  10. 4
  11. 4338
  12. 6
  13. 5
  14. 6417
  15. 9
  16. 4
  17. 7
  18. 3402
  19. 3
  20. 1
  21. 2
  22. 3
  23. 1
  24. 1
  25. 5
  26. 256
  27. 7
  28. 744
  29. 4
  30. 2
  31. 1088
  32. 7
  33. 1
  34. 273
  35. 4842
  36. 3
  37. 444
  38. 456
  39. 2
  40. 1

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