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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 552 · 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.Which of these numbers is divisible by 6? (a) 1955 (b) 1944 (c) 1942 (d) 1945
  2. 2.What is the units digit of 4^43? (a) 5 (b) 4 (c) 3 (d) 6
  3. 3.What is the units digit of 9^56? (a) 2 (b) 1 (c) 0 (d) 9
  4. 4.Which of these numbers is divisible by 9? (a) 6339 (b) 6333 (c) 6321 (d) 6327
  5. 5.Which of these numbers is divisible by 8? (a) 509 (b) 518 (c) 504 (d) 501
  6. 6.What is the units digit of 4^24? (a) 5 (b) 4 (c) 6 (d) 7
  7. 7.What is the smallest number you must add to 73 to make it divisible by 3? (a) 1 (b) 2
  8. 8.Which of these numbers is divisible by 4? (a) 3391 (b) 3394 (c) 3388 (d) 3387
  9. 9.What is the units digit of 4^31? (a) 6 (b) 5 (c) 4 (d) 3
  10. 10.What is the units digit of 2^49? (a) 6 (b) 4 (c) 8 (d) 2
  11. 11.What is the smallest number you must add to 357 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
  12. 12.What is the units digit of 9^59? (a) 7 (b) 8 (c) 9 (d) 1
  13. 13.What is the remainder when 127 is divided by 3? (a) 1 (b) 0 (c) 2
  14. 14.What is the units digit of 7^14? (a) 3 (b) 9 (c) 7 (d) 1
  15. 15.What is the smallest number you must add to 736 to make it divisible by 12? (a) 9 (b) 7 (c) 8 (d) 4
  16. 16.What is the smallest number you must add to 630 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
  17. 17.What is the remainder when 244 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  18. 18.Which of these numbers is divisible by 4? (a) 924 (b) 923 (c) 931 (d) 926
  19. 19.What is the smallest number you must add to 213 to make it divisible by 10? (a) 6 (b) 3 (c) 8 (d) 7
  20. 20.Which of these numbers is divisible by 9? (a) 2367 (b) 2378 (c) 2359 (d) 2369
  21. 21.What is the smallest number you must add to 189 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  22. 22.What is the units digit of 3^42? (a) 1 (b) 9 (c) 3 (d) 7
  23. 23.What is the remainder when 302 is divided by 8? (a) 2 (b) 7 (c) 5 (d) 6
  24. 24.Which of these numbers is divisible by 9? (a) 6166 (b) 6156 (c) 6149 (d) 6163
  25. 25.What is the smallest number you must add to 275 to make it divisible by 9? (a) 6 (b) 5 (c) 4 (d) 3
  26. 26.What is the remainder when 158 is divided by 12? (a) 2 (b) 3 (c) 10 (d) 1
  27. 27.What is the smallest number you must add to 362 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  28. 28.What is the smallest number you must add to 657 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
  29. 29.What is the units digit of 3^52? (a) 3 (b) 9 (c) 7 (d) 1
  30. 30.What is the smallest number you must add to 629 to make it divisible by 11? (a) 10 (b) 2 (c) 8 (d) 9
  31. 31.What is the remainder when 284 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  32. 32.What is the smallest number you must add to 194 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
  33. 33.What is the units digit of 7^54? (a) 1 (b) 9 (c) 7 (d) 3
  34. 34.What is the units digit of 9^16? (a) 1 (b) 0 (c) 2 (d) 9
  35. 35.What is the smallest number you must add to 370 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 2
  36. 36.Which of these numbers is divisible by 9? (a) 2481 (b) 2489 (c) 2467 (d) 2475
  37. 37.What is the smallest number you must add to 340 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
  38. 38.Which of these numbers is divisible by 6? (a) 279 (b) 273 (c) 270 (d) 267
  39. 39.What is the remainder when 486 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
  40. 40.Which of these numbers is divisible by 11? (a) 7675 (b) 7655 (c) 7656 (d) 7662

Answer key

  1. 1944
  2. 4
  3. 1
  4. 6327
  5. 504
  6. 6
  7. 2
  8. 3388
  9. 4
  10. 2
  11. 3
  12. 9
  13. 1
  14. 9
  15. 8
  16. 8
  17. 4
  18. 924
  19. 7
  20. 2367
  21. 3
  22. 9
  23. 6
  24. 6156
  25. 4
  26. 2
  27. 3
  28. 3
  29. 1
  30. 9
  31. 4
  32. 1
  33. 9
  34. 1
  35. 2
  36. 2475
  37. 4
  38. 270
  39. 6
  40. 7656

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