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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 63 · 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 3^31? (a) 1 (b) 7 (c) 9 (d) 3
  2. 2.What is the remainder when 134 is divided by 3? (a) 2 (b) 1 (c) 0
  3. 3.What is the remainder when 359 is divided by 9? (a) 1 (b) 7 (c) 6 (d) 8
  4. 4.What is the smallest number you must add to 461 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 2
  5. 5.What is the smallest number you must add to 462 to make it divisible by 10? (a) 8 (b) 2 (c) 9 (d) 7
  6. 6.What is the units digit of 7^10? (a) 1 (b) 3 (c) 7 (d) 9
  7. 7.Which of these numbers is divisible by 9? (a) 2421 (b) 2419 (c) 2433 (d) 2429
  8. 8.What is the smallest number you must add to 503 to make it divisible by 11? (a) 4 (b) 3 (c) 8 (d) 2
  9. 9.What is the remainder when 61 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
  10. 10.What is the units digit of 3^48? (a) 3 (b) 1 (c) 7 (d) 9
  11. 11.Which of these numbers is divisible by 11? (a) 4516 (b) 4510 (c) 4523 (d) 4505
  12. 12.What is the units digit of 8^42? (a) 6 (b) 2 (c) 8 (d) 4
  13. 13.What is the remainder when 74 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  14. 14.What is the units digit of 4^20? (a) 5 (b) 6 (c) 4 (d) 7
  15. 15.What is the smallest number you must add to 56 to make it divisible by 3? (a) 1 (b) 2
  16. 16.What is the smallest number you must add to 241 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
  17. 17.Which of these numbers is divisible by 4? (a) 1315 (b) 1316 (c) 1317 (d) 1322
  18. 18.What is the smallest number you must add to 258 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
  19. 19.What is the remainder when 291 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  20. 20.What is the smallest number you must add to 49 to make it divisible by 3? (a) 1 (b) 2
  21. 21.What is the smallest number you must add to 178 to make it divisible by 3? (a) 1 (b) 2
  22. 22.Which of these numbers is divisible by 8? (a) 1424 (b) 1430 (c) 1423 (d) 1434
  23. 23.Which of these numbers is divisible by 3? (a) 299 (b) 301 (c) 300 (d) 305
  24. 24.What is the units digit of 7^14? (a) 1 (b) 9 (c) 3 (d) 7
  25. 25.What is the remainder when 199 is divided by 9? (a) 2 (b) 8 (c) 1 (d) 0
  26. 26.What is the units digit of 2^20? (a) 4 (b) 2 (c) 8 (d) 6
  27. 27.What is the units digit of 4^23? (a) 6 (b) 5 (c) 3 (d) 4
  28. 28.What is the remainder when 43 is divided by 3? (a) 0 (b) 2 (c) 1
  29. 29.What is the remainder when 275 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
  30. 30.What is the remainder when 218 is divided by 7? (a) 0 (b) 2 (c) 1 (d) 6
  31. 31.What is the units digit of 2^13? (a) 8 (b) 2 (c) 4 (d) 6
  32. 32.What is the smallest number you must add to 206 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  33. 33.What is the smallest number you must add to 212 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
  34. 34.What is the units digit of 8^38? (a) 4 (b) 6 (c) 8 (d) 2
  35. 35.What is the units digit of 3^56? (a) 9 (b) 7 (c) 3 (d) 1
  36. 36.What is the remainder when 268 is divided by 9? (a) 6 (b) 2 (c) 7 (d) 8
  37. 37.What is the units digit of 9^32? (a) 9 (b) 0 (c) 1 (d) 2
  38. 38.What is the units digit of 4^42? (a) 4 (b) 5 (c) 6 (d) 7
  39. 39.Which of these numbers is divisible by 8? (a) 361 (b) 354 (c) 374 (d) 360
  40. 40.What is the remainder when 178 is divided by 3? (a) 0 (b) 1 (c) 2

Answer key

  1. 7
  2. 2
  3. 8
  4. 1
  5. 8
  6. 9
  7. 2421
  8. 3
  9. 1
  10. 1
  11. 4510
  12. 4
  13. 2
  14. 6
  15. 1
  16. 2
  17. 1316
  18. 2
  19. 3
  20. 2
  21. 2
  22. 1424
  23. 300
  24. 9
  25. 1
  26. 6
  27. 4
  28. 1
  29. 3
  30. 1
  31. 2
  32. 4
  33. 4
  34. 4
  35. 1
  36. 7
  37. 1
  38. 6
  39. 360
  40. 1

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