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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 905 · 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 remainder when 587 is divided by 10? (a) 7 (b) 8 (c) 3 (d) 6
  2. 2.What is the remainder when 520 is divided by 9? (a) 7 (b) 6 (c) 2 (d) 8
  3. 3.What is the units digit of 8^17? (a) 6 (b) 4 (c) 8 (d) 2
  4. 4.Which of these numbers is divisible by 4? (a) 714 (b) 722 (c) 717 (d) 716
  5. 5.What is the remainder when 218 is divided by 10? (a) 8 (b) 7 (c) 2 (d) 9
  6. 6.What is the smallest number you must add to 134 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
  7. 7.Which of these numbers is divisible by 4? (a) 1093 (b) 1101 (c) 1096 (d) 1097
  8. 8.What is the smallest number you must add to 67 to make it divisible by 3? (a) 2 (b) 1
  9. 9.Which of these numbers is divisible by 3? (a) 1179 (b) 1184 (c) 1181 (d) 1178
  10. 10.What is the smallest number you must add to 265 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
  11. 11.What is the smallest number you must add to 277 to make it divisible by 8? (a) 2 (b) 5 (c) 4 (d) 3
  12. 12.What is the units digit of 2^26? (a) 4 (b) 8 (c) 6 (d) 2
  13. 13.Which of these numbers is divisible by 4? (a) 227 (b) 230 (c) 221 (d) 224
  14. 14.Which of these numbers is divisible by 3? (a) 710 (b) 708 (c) 712 (d) 707
  15. 15.What is the smallest number you must add to 379 to make it divisible by 12? (a) 4 (b) 5 (c) 7 (d) 6
  16. 16.What is the units digit of 8^40? (a) 8 (b) 6 (c) 4 (d) 2
  17. 17.What is the remainder when 244 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
  18. 18.What is the units digit of 8^14? (a) 4 (b) 2 (c) 8 (d) 6
  19. 19.What is the smallest number you must add to 835 to make it divisible by 12? (a) 6 (b) 4 (c) 7 (d) 5
  20. 20.What is the smallest number you must add to 237 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 2
  21. 21.Which of these numbers is divisible by 6? (a) 258 (b) 254 (c) 267 (d) 262
  22. 22.Which of these numbers is divisible by 3? (a) 532 (b) 529 (c) 527 (d) 528
  23. 23.What is the remainder when 352 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
  24. 24.Which of these numbers is divisible by 8? (a) 4198 (b) 4200 (c) 4204 (d) 4209
  25. 25.What is the smallest number you must add to 442 to make it divisible by 8? (a) 2 (b) 5 (c) 7 (d) 6
  26. 26.What is the units digit of 2^23? (a) 2 (b) 6 (c) 8 (d) 4
  27. 27.What is the smallest number you must add to 570 to make it divisible by 12? (a) 5 (b) 8 (c) 7 (d) 6
  28. 28.Which of these numbers is divisible by 8? (a) 6302 (b) 6318 (c) 6305 (d) 6304
  29. 29.What is the remainder when 338 is divided by 9? (a) 6 (b) 4 (c) 5 (d) 7
  30. 30.What is the units digit of 8^54? (a) 8 (b) 2 (c) 4 (d) 6
  31. 31.What is the smallest number you must add to 94 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 4
  32. 32.What is the remainder when 226 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
  33. 33.What is the smallest number you must add to 475 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
  34. 34.What is the smallest number you must add to 695 to make it divisible by 11? (a) 2 (b) 9 (c) 10 (d) 8
  35. 35.What is the units digit of 3^31? (a) 9 (b) 1 (c) 7 (d) 3
  36. 36.What is the units digit of 4^12? (a) 4 (b) 5 (c) 6 (d) 7
  37. 37.What is the smallest number you must add to 735 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
  38. 38.What is the remainder when 182 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
  39. 39.What is the smallest number you must add to 515 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 7
  40. 40.What is the smallest number you must add to 195 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5

Answer key

  1. 7
  2. 7
  3. 8
  4. 716
  5. 8
  6. 4
  7. 1096
  8. 2
  9. 1179
  10. 5
  11. 3
  12. 4
  13. 224
  14. 708
  15. 5
  16. 6
  17. 4
  18. 4
  19. 5
  20. 3
  21. 258
  22. 528
  23. 2
  24. 4200
  25. 6
  26. 8
  27. 6
  28. 6304
  29. 5
  30. 4
  31. 5
  32. 2
  33. 5
  34. 9
  35. 7
  36. 6
  37. 5
  38. 2
  39. 5
  40. 5

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