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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 5 · 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 686 to make it divisible by 11? (a) 7 (b) 4 (c) 6 (d) 8
  2. 2.What is the remainder when 267 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 3
  3. 3.Which of these numbers is divisible by 6? (a) 1143 (b) 1146 (c) 1153 (d) 1149
  4. 4.Which of these numbers is divisible by 4? (a) 2377 (b) 2383 (c) 2376 (d) 2375
  5. 5.What is the smallest number you must add to 208 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 4
  6. 6.What is the smallest number you must add to 148 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
  7. 7.Which of these numbers is divisible by 9? (a) 2629 (b) 2644 (c) 2626 (d) 2628
  8. 8.What is the smallest number you must add to 164 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
  9. 9.What is the remainder when 141 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
  10. 10.What is the units digit of 4^27? (a) 6 (b) 5 (c) 3 (d) 4
  11. 11.Which of these numbers is divisible by 6? (a) 983 (b) 985 (c) 994 (d) 984
  12. 12.What is the remainder when 330 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
  13. 13.What is the units digit of 9^45? (a) 1 (b) 7 (c) 8 (d) 9
  14. 14.Which of these numbers is divisible by 9? (a) 6855 (b) 6833 (c) 6841 (d) 6840
  15. 15.What is the remainder when 123 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
  16. 16.What is the smallest number you must add to 211 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 1
  17. 17.What is the smallest number you must add to 158 to make it divisible by 3? (a) 2 (b) 1
  18. 18.What is the remainder when 253 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
  19. 19.What is the smallest number you must add to 329 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  20. 20.Which of these numbers is divisible by 3? (a) 2687 (b) 2689 (c) 2692 (d) 2688
  21. 21.What is the smallest number you must add to 582 to make it divisible by 8? (a) 1 (b) 2 (c) 3 (d) 6
  22. 22.What is the units digit of 4^24? (a) 7 (b) 6 (c) 4 (d) 5
  23. 23.Which of these numbers is divisible by 9? (a) 7611 (b) 7614 (c) 7621 (d) 7625
  24. 24.Which of these numbers is divisible by 3? (a) 695 (b) 698 (c) 691 (d) 693
  25. 25.What is the smallest number you must add to 130 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  26. 26.What is the smallest number you must add to 131 to make it divisible by 12? (a) 11 (b) 1 (c) 2 (d) 3
  27. 27.What is the smallest number you must add to 371 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  28. 28.What is the smallest number you must add to 466 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 2
  29. 29.What is the smallest number you must add to 273 to make it divisible by 9? (a) 5 (b) 3 (c) 7 (d) 6
  30. 30.Which of these numbers is divisible by 3? (a) 1059 (b) 1063 (c) 1061 (d) 1057
  31. 31.What is the remainder when 179 is divided by 3? (a) 2 (b) 0 (c) 1
  32. 32.What is the units digit of 7^10? (a) 9 (b) 7 (c) 1 (d) 3
  33. 33.What is the remainder when 184 is divided by 3? (a) 1 (b) 2 (c) 0
  34. 34.What is the units digit of 3^51? (a) 3 (b) 9 (c) 1 (d) 7
  35. 35.Which of these numbers is divisible by 11? (a) 4678 (b) 4671 (c) 4675 (d) 4696
  36. 36.Which of these numbers is divisible by 8? (a) 2057 (b) 2067 (c) 2064 (d) 2076
  37. 37.What is the units digit of 2^11? (a) 2 (b) 8 (c) 6 (d) 4
  38. 38.What is the smallest number you must add to 191 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  39. 39.What is the smallest number you must add to 423 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
  40. 40.Which of these numbers is divisible by 8? (a) 3768 (b) 3766 (c) 3783 (d) 3773

Answer key

  1. 7
  2. 6
  3. 1146
  4. 2376
  5. 2
  6. 6
  7. 2628
  8. 4
  9. 1
  10. 4
  11. 984
  12. 1
  13. 9
  14. 6840
  15. 3
  16. 5
  17. 1
  18. 5
  19. 1
  20. 2688
  21. 2
  22. 6
  23. 7614
  24. 693
  25. 2
  26. 1
  27. 4
  28. 6
  29. 6
  30. 1059
  31. 2
  32. 9
  33. 1
  34. 7
  35. 4675
  36. 2064
  37. 8
  38. 4
  39. 1
  40. 3768

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