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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 611 · 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 units digit of 9^44? (a) 1 (b) 9 (c) 2 (d) 0
  2. 2.What is the smallest number you must add to 187 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  3. 3.What is the smallest number you must add to 157 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  4. 4.What is the remainder when 64 is divided by 3? (a) 2 (b) 1 (c) 0
  5. 5.What is the units digit of 7^20? (a) 9 (b) 7 (c) 1 (d) 3
  6. 6.What is the smallest number you must add to 160 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 4
  7. 7.What is the units digit of 9^17? (a) 8 (b) 1 (c) 7 (d) 9
  8. 8.What is the remainder when 97 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
  9. 9.What is the smallest number you must add to 627 to make it divisible by 9? (a) 2 (b) 4 (c) 6 (d) 3
  10. 10.What is the remainder when 70 is divided by 3? (a) 1 (b) 0 (c) 2
  11. 11.What is the smallest number you must add to 286 to make it divisible by 9? (a) 3 (b) 2 (c) 7 (d) 1
  12. 12.What is the units digit of 3^10? (a) 3 (b) 9 (c) 1 (d) 7
  13. 13.What is the smallest number you must add to 113 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 1
  14. 14.What is the smallest number you must add to 219 to make it divisible by 12? (a) 8 (b) 3 (c) 9 (d) 10
  15. 15.What is the units digit of 3^12? (a) 9 (b) 7 (c) 3 (d) 1
  16. 16.What is the smallest number you must add to 460 to make it divisible by 9? (a) 8 (b) 1 (c) 6 (d) 7
  17. 17.What is the remainder when 590 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 7
  18. 18.What is the smallest number you must add to 575 to make it divisible by 9? (a) 2 (b) 8 (c) 1 (d) 3
  19. 19.What is the units digit of 2^17? (a) 8 (b) 4 (c) 2 (d) 6
  20. 20.What is the units digit of 2^30? (a) 8 (b) 2 (c) 4 (d) 6
  21. 21.What is the smallest number you must add to 454 to make it divisible by 7? (a) 1 (b) 6 (c) 2 (d) 3
  22. 22.What is the smallest number you must add to 256 to make it divisible by 11? (a) 9 (b) 7 (c) 8 (d) 3
  23. 23.What is the smallest number you must add to 535 to make it divisible by 12? (a) 7 (b) 4 (c) 5 (d) 6
  24. 24.Which of these numbers is divisible by 11? (a) 9570 (b) 9587 (c) 9569 (d) 9571
  25. 25.Which of these numbers is divisible by 4? (a) 2946 (b) 2940 (c) 2938 (d) 2941
  26. 26.What is the units digit of 2^36? (a) 4 (b) 6 (c) 8 (d) 2
  27. 27.Which of these numbers is divisible by 11? (a) 1227 (b) 1235 (c) 1232 (d) 1251
  28. 28.What is the smallest number you must add to 235 to make it divisible by 8? (a) 4 (b) 5 (c) 6 (d) 3
  29. 29.What is the remainder when 508 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
  30. 30.Which of these numbers is divisible by 9? (a) 6944 (b) 6948 (c) 6952 (d) 6960
  31. 31.Which of these numbers is divisible by 3? (a) 702 (b) 700 (c) 704 (d) 707
  32. 32.What is the remainder when 43 is divided by 3? (a) 2 (b) 1 (c) 0
  33. 33.Which of these numbers is divisible by 3? (a) 1258 (b) 1262 (c) 1265 (d) 1260
  34. 34.What is the units digit of 3^47? (a) 9 (b) 7 (c) 1 (d) 3
  35. 35.What is the units digit of 2^23? (a) 2 (b) 6 (c) 8 (d) 4
  36. 36.What is the units digit of 9^18? (a) 0 (b) 1 (c) 2 (d) 9
  37. 37.Which of these numbers is divisible by 8? (a) 2507 (b) 2502 (c) 2491 (d) 2496
  38. 38.What is the smallest number you must add to 236 to make it divisible by 3? (a) 2 (b) 1
  39. 39.Which of these numbers is divisible by 4? (a) 677 (b) 680 (c) 681 (d) 685
  40. 40.What is the smallest number you must add to 112 to make it divisible by 10? (a) 9 (b) 8 (c) 2 (d) 7

Answer key

  1. 1
  2. 1
  3. 3
  4. 1
  5. 1
  6. 2
  7. 9
  8. 2
  9. 3
  10. 1
  11. 2
  12. 9
  13. 7
  14. 9
  15. 1
  16. 8
  17. 5
  18. 1
  19. 2
  20. 4
  21. 1
  22. 8
  23. 5
  24. 9570
  25. 2940
  26. 6
  27. 1232
  28. 5
  29. 4
  30. 6948
  31. 702
  32. 1
  33. 1260
  34. 7
  35. 8
  36. 1
  37. 2496
  38. 1
  39. 680
  40. 8

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