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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 719 · 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 3^45? (a) 1 (b) 3 (c) 7 (d) 9
  2. 2.Which of these numbers is divisible by 3? (a) 663 (b) 665 (c) 668 (d) 661
  3. 3.Which of these numbers is divisible by 4? (a) 1159 (b) 1165 (c) 1162 (d) 1160
  4. 4.What is the smallest number you must add to 161 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 5
  5. 5.What is the remainder when 378 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
  6. 6.What is the smallest number you must add to 270 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  7. 7.What is the smallest number you must add to 790 to make it divisible by 11? (a) 9 (b) 1 (c) 2 (d) 3
  8. 8.What is the units digit of 3^41? (a) 9 (b) 3 (c) 7 (d) 1
  9. 9.Which of these numbers is divisible by 3? (a) 1376 (b) 1372 (c) 1371 (d) 1370
  10. 10.What is the remainder when 127 is divided by 6? (a) 1 (b) 5 (c) 2 (d) 0
  11. 11.What is the remainder when 254 is divided by 9? (a) 2 (b) 1 (c) 3 (d) 7
  12. 12.What is the remainder when 249 is divided by 12? (a) 10 (b) 3 (c) 9 (d) 8
  13. 13.What is the smallest number you must add to 108 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  14. 14.What is the units digit of 8^49? (a) 4 (b) 6 (c) 8 (d) 2
  15. 15.What is the units digit of 7^50? (a) 7 (b) 9 (c) 3 (d) 1
  16. 16.What is the units digit of 3^42? (a) 7 (b) 1 (c) 3 (d) 9
  17. 17.What is the remainder when 314 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
  18. 18.What is the remainder when 215 is divided by 8? (a) 1 (b) 7 (c) 6 (d) 5
  19. 19.What is the units digit of 2^48? (a) 2 (b) 6 (c) 8 (d) 4
  20. 20.Which of these numbers is divisible by 8? (a) 6446 (b) 6453 (c) 6436 (d) 6440
  21. 21.Which of these numbers is divisible by 8? (a) 3317 (b) 3304 (c) 3300 (d) 3310
  22. 22.What is the smallest number you must add to 434 to make it divisible by 9? (a) 2 (b) 6 (c) 8 (d) 7
  23. 23.What is the smallest number you must add to 229 to make it divisible by 7? (a) 1 (b) 5 (c) 3 (d) 2
  24. 24.What is the smallest number you must add to 47 to make it divisible by 3? (a) 1 (b) 2
  25. 25.Which of these numbers is divisible by 6? (a) 3433 (b) 3442 (c) 3432 (d) 3427
  26. 26.What is the smallest number you must add to 792 to make it divisible by 10? (a) 7 (b) 9 (c) 2 (d) 8
  27. 27.What is the remainder when 794 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
  28. 28.What is the units digit of 8^34? (a) 4 (b) 8 (c) 2 (d) 6
  29. 29.What is the remainder when 359 is divided by 8? (a) 6 (b) 5 (c) 7 (d) 1
  30. 30.What is the remainder when 644 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5
  31. 31.What is the remainder when 259 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
  32. 32.Which of these numbers is divisible by 4? (a) 2727 (b) 2721 (c) 2724 (d) 2729
  33. 33.What is the smallest number you must add to 354 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  34. 34.What is the smallest number you must add to 400 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
  35. 35.What is the units digit of 9^10? (a) 9 (b) 0 (c) 1 (d) 2
  36. 36.Which of these numbers is divisible by 6? (a) 2345 (b) 2347 (c) 2337 (d) 2340
  37. 37.What is the smallest number you must add to 203 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 2
  38. 38.Which of these numbers is divisible by 4? (a) 1684 (b) 1689 (c) 1685 (d) 1682
  39. 39.What is the smallest number you must add to 185 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 3
  40. 40.What is the smallest number you must add to 376 to make it divisible by 9? (a) 1 (b) 3 (c) 2 (d) 7

Answer key

  1. 3
  2. 663
  3. 1160
  4. 1
  5. 2
  6. 2
  7. 2
  8. 3
  9. 1371
  10. 1
  11. 2
  12. 9
  13. 2
  14. 8
  15. 9
  16. 9
  17. 4
  18. 7
  19. 6
  20. 6440
  21. 3304
  22. 7
  23. 2
  24. 1
  25. 3432
  26. 8
  27. 2
  28. 4
  29. 7
  30. 4
  31. 1
  32. 2724
  33. 1
  34. 8
  35. 1
  36. 2340
  37. 1
  38. 1684
  39. 4
  40. 2

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