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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 879 · 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 2^53? (a) 6 (b) 8 (c) 4 (d) 2
  2. 2.What is the smallest number you must add to 560 to make it divisible by 9? (a) 8 (b) 7 (c) 2 (d) 6
  3. 3.Which of these numbers is divisible by 4? (a) 304 (b) 309 (c) 307 (d) 303
  4. 4.What is the remainder when 256 is divided by 9? (a) 6 (b) 4 (c) 3 (d) 5
  5. 5.What is the smallest number you must add to 247 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 4
  6. 6.What is the smallest number you must add to 426 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 2
  7. 7.Which of these numbers is divisible by 4? (a) 2843 (b) 2851 (c) 2845 (d) 2844
  8. 8.Which of these numbers is divisible by 9? (a) 6934 (b) 6921 (c) 6926 (d) 6913
  9. 9.Which of these numbers is divisible by 8? (a) 418 (b) 414 (c) 408 (d) 407
  10. 10.What is the smallest number you must add to 308 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
  11. 11.What is the smallest number you must add to 134 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  12. 12.What is the units digit of 4^55? (a) 3 (b) 4 (c) 6 (d) 5
  13. 13.What is the smallest number you must add to 375 to make it divisible by 12? (a) 10 (b) 3 (c) 9 (d) 8
  14. 14.What is the units digit of 7^30? (a) 1 (b) 7 (c) 9 (d) 3
  15. 15.What is the units digit of 2^46? (a) 2 (b) 4 (c) 8 (d) 6
  16. 16.What is the units digit of 3^46? (a) 9 (b) 3 (c) 1 (d) 7
  17. 17.Which of these numbers is divisible by 3? (a) 1469 (b) 1472 (c) 1470 (d) 1474
  18. 18.What is the units digit of 4^24? (a) 7 (b) 6 (c) 5 (d) 4
  19. 19.What is the units digit of 4^11? (a) 5 (b) 3 (c) 4 (d) 6
  20. 20.What is the remainder when 237 is divided by 7? (a) 1 (b) 6 (c) 5 (d) 4
  21. 21.What is the smallest number you must add to 149 to make it divisible by 7? (a) 2 (b) 5 (c) 6 (d) 4
  22. 22.Which of these numbers is divisible by 11? (a) 2617 (b) 2631 (c) 2624 (d) 2618
  23. 23.What is the units digit of 9^49? (a) 7 (b) 9 (c) 8 (d) 1
  24. 24.What is the smallest number you must add to 593 to make it divisible by 9? (a) 8 (b) 3 (c) 1 (d) 2
  25. 25.Which of these numbers is divisible by 11? (a) 1204 (b) 1188 (c) 1179 (d) 1191
  26. 26.What is the smallest number you must add to 100 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
  27. 27.What is the smallest number you must add to 167 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  28. 28.What is the units digit of 3^47? (a) 9 (b) 3 (c) 1 (d) 7
  29. 29.Which of these numbers is divisible by 11? (a) 7222 (b) 7239 (c) 7229 (d) 7227
  30. 30.What is the units digit of 3^36? (a) 7 (b) 1 (c) 9 (d) 3
  31. 31.What is the units digit of 7^44? (a) 7 (b) 3 (c) 1 (d) 9
  32. 32.Which of these numbers is divisible by 8? (a) 6830 (b) 6843 (c) 6836 (d) 6832
  33. 33.What is the units digit of 3^40? (a) 3 (b) 9 (c) 7 (d) 1
  34. 34.What is the smallest number you must add to 689 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 3
  35. 35.Which of these numbers is divisible by 11? (a) 8107 (b) 8102 (c) 8125 (d) 8112
  36. 36.What is the units digit of 8^28? (a) 4 (b) 2 (c) 6 (d) 8
  37. 37.What is the smallest number you must add to 920 to make it divisible by 12? (a) 5 (b) 8 (c) 4 (d) 3
  38. 38.What is the smallest number you must add to 369 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  39. 39.What is the smallest number you must add to 602 to make it divisible by 10? (a) 2 (b) 9 (c) 8 (d) 7
  40. 40.What is the smallest number you must add to 225 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4

Answer key

  1. 2
  2. 7
  3. 304
  4. 4
  5. 5
  6. 6
  7. 2844
  8. 6921
  9. 408
  10. 4
  11. 2
  12. 4
  13. 9
  14. 9
  15. 4
  16. 9
  17. 1470
  18. 6
  19. 4
  20. 6
  21. 5
  22. 2618
  23. 9
  24. 1
  25. 1188
  26. 5
  27. 3
  28. 7
  29. 7227
  30. 1
  31. 1
  32. 6832
  33. 1
  34. 4
  35. 8107
  36. 6
  37. 4
  38. 1
  39. 8
  40. 5

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