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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 814 · 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 439 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
  2. 2.What is the remainder when 226 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
  3. 3.What is the units digit of 8^47? (a) 8 (b) 4 (c) 6 (d) 2
  4. 4.What is the remainder when 43 is divided by 3? (a) 2 (b) 1 (c) 0
  5. 5.Which of these numbers is divisible by 4? (a) 3150 (b) 3144 (c) 3141 (d) 3147
  6. 6.What is the smallest number you must add to 337 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  7. 7.What is the units digit of 8^35? (a) 2 (b) 6 (c) 8 (d) 4
  8. 8.What is the smallest number you must add to 129 to make it divisible by 10? (a) 1 (b) 3 (c) 2 (d) 9
  9. 9.What is the remainder when 339 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 4
  10. 10.Which of these numbers is divisible by 8? (a) 2755 (b) 2748 (c) 2744 (d) 2737
  11. 11.What is the units digit of 2^44? (a) 8 (b) 6 (c) 4 (d) 2
  12. 12.Which of these numbers is divisible by 8? (a) 1811 (b) 1803 (c) 1808 (d) 1821
  13. 13.What is the units digit of 8^31? (a) 4 (b) 8 (c) 2 (d) 6
  14. 14.What is the units digit of 2^48? (a) 8 (b) 4 (c) 6 (d) 2
  15. 15.What is the smallest number you must add to 331 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  16. 16.Which of these numbers is divisible by 3? (a) 323 (b) 324 (c) 328 (d) 325
  17. 17.What is the remainder when 127 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  18. 18.What is the smallest number you must add to 333 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 2
  19. 19.What is the smallest number you must add to 113 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 5
  20. 20.What is the smallest number you must add to 193 to make it divisible by 3? (a) 1 (b) 2
  21. 21.What is the units digit of 9^60? (a) 9 (b) 0 (c) 1 (d) 2
  22. 22.Which of these numbers is divisible by 11? (a) 6949 (b) 6941 (c) 6939 (d) 6954
  23. 23.What is the units digit of 9^45? (a) 1 (b) 9 (c) 8 (d) 7
  24. 24.What is the remainder when 472 is divided by 7? (a) 4 (b) 2 (c) 3 (d) 5
  25. 25.Which of these numbers is divisible by 3? (a) 1325 (b) 1318 (c) 1320 (d) 1321
  26. 26.Which of these numbers is divisible by 11? (a) 4671 (b) 4689 (c) 4675 (d) 4681
  27. 27.What is the remainder when 283 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  28. 28.What is the units digit of 2^27? (a) 2 (b) 6 (c) 4 (d) 8
  29. 29.What is the remainder when 160 is divided by 11? (a) 8 (b) 7 (c) 6 (d) 5
  30. 30.What is the smallest number you must add to 366 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  31. 31.What is the smallest number you must add to 505 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 1
  32. 32.What is the units digit of 8^38? (a) 2 (b) 8 (c) 6 (d) 4
  33. 33.Which of these numbers is divisible by 4? (a) 2638 (b) 2635 (c) 2632 (d) 2630
  34. 34.What is the units digit of 8^18? (a) 8 (b) 2 (c) 6 (d) 4
  35. 35.What is the remainder when 443 is divided by 10? (a) 3 (b) 7 (c) 4 (d) 2
  36. 36.What is the remainder when 556 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 7
  37. 37.What is the smallest number you must add to 718 to make it divisible by 11? (a) 3 (b) 7 (c) 9 (d) 8
  38. 38.Which of these numbers is divisible by 9? (a) 6739 (b) 6718 (c) 6723 (d) 6724
  39. 39.What is the smallest number you must add to 687 to make it divisible by 9? (a) 7 (b) 3 (c) 5 (d) 6
  40. 40.What is the smallest number you must add to 358 to make it divisible by 9? (a) 7 (b) 3 (c) 1 (d) 2

Answer key

  1. 5
  2. 2
  3. 2
  4. 1
  5. 3144
  6. 3
  7. 2
  8. 1
  9. 3
  10. 2744
  11. 6
  12. 1808
  13. 2
  14. 6
  15. 4
  16. 324
  17. 2
  18. 3
  19. 4
  20. 2
  21. 1
  22. 6941
  23. 9
  24. 3
  25. 1320
  26. 4675
  27. 3
  28. 8
  29. 6
  30. 4
  31. 11
  32. 4
  33. 2632
  34. 4
  35. 3
  36. 6
  37. 8
  38. 6723
  39. 6
  40. 2

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