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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 641 · 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 403 to make it divisible by 11? (a) 4 (b) 7 (c) 3 (d) 5
  2. 2.What is the smallest number you must add to 667 to make it divisible by 11? (a) 5 (b) 4 (c) 3 (d) 7
  3. 3.What is the units digit of 8^42? (a) 2 (b) 6 (c) 4 (d) 8
  4. 4.What is the smallest number you must add to 86 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
  5. 5.Which of these numbers is divisible by 8? (a) 4899 (b) 4886 (c) 4888 (d) 4890
  6. 6.What is the units digit of 2^21? (a) 6 (b) 2 (c) 8 (d) 4
  7. 7.What is the smallest number you must add to 267 to make it divisible by 10? (a) 3 (b) 2 (c) 7 (d) 4
  8. 8.What is the remainder when 638 is divided by 9? (a) 1 (b) 6 (c) 7 (d) 8
  9. 9.What is the remainder when 403 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
  10. 10.What is the units digit of 3^13? (a) 7 (b) 9 (c) 1 (d) 3
  11. 11.What is the smallest number you must add to 530 to make it divisible by 9? (a) 2 (b) 1 (c) 8 (d) 3
  12. 12.What is the smallest number you must add to 141 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  13. 13.Which of these numbers is divisible by 4? (a) 2654 (b) 2648 (c) 2649 (d) 2646
  14. 14.Which of these numbers is divisible by 8? (a) 2596 (b) 2611 (c) 2600 (d) 2606
  15. 15.What is the units digit of 2^49? (a) 4 (b) 8 (c) 6 (d) 2
  16. 16.Which of these numbers is divisible by 4? (a) 2179 (b) 2174 (c) 2172 (d) 2171
  17. 17.What is the smallest number you must add to 268 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
  18. 18.What is the smallest number you must add to 555 to make it divisible by 9? (a) 3 (b) 2 (c) 6 (d) 4
  19. 19.What is the units digit of 4^21? (a) 5 (b) 6 (c) 3 (d) 4
  20. 20.Which of these numbers is divisible by 8? (a) 2685 (b) 2691 (c) 2688 (d) 2699
  21. 21.What is the smallest number you must add to 314 to make it divisible by 11? (a) 7 (b) 6 (c) 5 (d) 4
  22. 22.What is the units digit of 2^27? (a) 6 (b) 8 (c) 4 (d) 2
  23. 23.What is the smallest number you must add to 362 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  24. 24.Which of these numbers is divisible by 11? (a) 7367 (b) 7370 (c) 7377 (d) 7382
  25. 25.What is the smallest number you must add to 311 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 5
  26. 26.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
  27. 27.What is the remainder when 743 is divided by 11? (a) 6 (b) 8 (c) 7 (d) 5
  28. 28.What is the remainder when 380 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 8
  29. 29.What is the remainder when 103 is divided by 3? (a) 0 (b) 2 (c) 1
  30. 30.What is the remainder when 486 is divided by 11? (a) 9 (b) 2 (c) 1 (d) 3
  31. 31.Which of these numbers is divisible by 9? (a) 6164 (b) 6165 (c) 6176 (d) 6171
  32. 32.What is the remainder when 106 is divided by 3? (a) 2 (b) 0 (c) 1
  33. 33.What is the smallest number you must add to 65 to make it divisible by 3? (a) 2 (b) 1
  34. 34.What is the smallest number you must add to 398 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5
  35. 35.What is the smallest number you must add to 296 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
  36. 36.Which of these numbers is divisible by 3? (a) 1292 (b) 1290 (c) 1289 (d) 1295
  37. 37.What is the units digit of 9^22? (a) 1 (b) 2 (c) 9 (d) 0
  38. 38.Which of these numbers is divisible by 3? (a) 2676 (b) 2674 (c) 2680 (d) 2678
  39. 39.Which of these numbers is divisible by 3? (a) 2288 (b) 2286 (c) 2291 (d) 2284
  40. 40.What is the units digit of 2^25? (a) 2 (b) 6 (c) 8 (d) 4

Answer key

  1. 4
  2. 4
  3. 4
  4. 5
  5. 4888
  6. 2
  7. 3
  8. 8
  9. 3
  10. 3
  11. 1
  12. 3
  13. 2648
  14. 2600
  15. 2
  16. 2172
  17. 5
  18. 3
  19. 4
  20. 2688
  21. 5
  22. 8
  23. 3
  24. 7370
  25. 1
  26. 1
  27. 6
  28. 6
  29. 1
  30. 2
  31. 6165
  32. 1
  33. 1
  34. 4
  35. 4
  36. 1290
  37. 1
  38. 2676
  39. 2286
  40. 2

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