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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 274 · 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 remainder when 352 is divided by 10? (a) 1 (b) 3 (c) 8 (d) 2
  2. 2.What is the units digit of 9^52? (a) 2 (b) 9 (c) 0 (d) 1
  3. 3.What is the smallest number you must add to 883 to make it divisible by 11? (a) 7 (b) 8 (c) 9 (d) 3
  4. 4.What is the units digit of 2^42? (a) 2 (b) 8 (c) 6 (d) 4
  5. 5.What is the remainder when 458 is divided by 9? (a) 6 (b) 7 (c) 1 (d) 8
  6. 6.What is the remainder when 251 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
  7. 7.What is the remainder when 287 is divided by 9? (a) 1 (b) 7 (c) 6 (d) 8
  8. 8.What is the smallest number you must add to 239 to make it divisible by 11? (a) 4 (b) 3 (c) 2 (d) 8
  9. 9.What is the remainder when 501 is divided by 12? (a) 9 (b) 8 (c) 3 (d) 10
  10. 10.What is the smallest number you must add to 619 to make it divisible by 12? (a) 6 (b) 4 (c) 7 (d) 5
  11. 11.What is the smallest number you must add to 491 to make it divisible by 7? (a) 1 (b) 6 (c) 5 (d) 4
  12. 12.What is the smallest number you must add to 284 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
  13. 13.Which of these numbers is divisible by 3? (a) 533 (b) 534 (c) 538 (d) 535
  14. 14.What is the units digit of 3^17? (a) 1 (b) 9 (c) 3 (d) 7
  15. 15.Which of these numbers is divisible by 6? (a) 1435 (b) 1434 (c) 1432 (d) 1444
  16. 16.What is the units digit of 9^43? (a) 1 (b) 8 (c) 7 (d) 9
  17. 17.Which of these numbers is divisible by 3? (a) 698 (b) 695 (c) 696 (d) 701
  18. 18.What is the remainder when 331 is divided by 7? (a) 1 (b) 5 (c) 2 (d) 3
  19. 19.Which of these numbers is divisible by 8? (a) 2228 (b) 2213 (c) 2221 (d) 2216
  20. 20.What is the remainder when 858 is divided by 12? (a) 5 (b) 7 (c) 6 (d) 8
  21. 21.What is the units digit of 8^49? (a) 2 (b) 8 (c) 4 (d) 6
  22. 22.What is the smallest number you must add to 265 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  23. 23.What is the remainder when 310 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
  24. 24.What is the remainder when 245 is divided by 10? (a) 7 (b) 6 (c) 4 (d) 5
  25. 25.What is the remainder when 683 is divided by 11? (a) 10 (b) 1 (c) 2 (d) 0
  26. 26.What is the remainder when 223 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
  27. 27.What is the remainder when 370 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
  28. 28.Which of these numbers is divisible by 9? (a) 3631 (b) 3642 (c) 3650 (d) 3636
  29. 29.What is the remainder when 598 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 4
  30. 30.What is the smallest number you must add to 138 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  31. 31.What is the units digit of 7^34? (a) 9 (b) 7 (c) 3 (d) 1
  32. 32.Which of these numbers is divisible by 3? (a) 2623 (b) 2627 (c) 2622 (d) 2621
  33. 33.Which of these numbers is divisible by 11? (a) 5313 (b) 5310 (c) 5314 (d) 5328
  34. 34.What is the remainder when 332 is divided by 7? (a) 2 (b) 3 (c) 4 (d) 5
  35. 35.What is the smallest number you must add to 246 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  36. 36.What is the units digit of 7^10? (a) 1 (b) 9 (c) 3 (d) 7
  37. 37.What is the smallest number you must add to 44 to make it divisible by 3? (a) 2 (b) 1
  38. 38.What is the units digit of 8^31? (a) 2 (b) 6 (c) 8 (d) 4
  39. 39.Which of these numbers is divisible by 4? (a) 1311 (b) 1307 (c) 1308 (d) 1314
  40. 40.What is the remainder when 250 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2

Answer key

  1. 2
  2. 1
  3. 8
  4. 4
  5. 8
  6. 1
  7. 8
  8. 3
  9. 9
  10. 5
  11. 6
  12. 4
  13. 534
  14. 3
  15. 1434
  16. 9
  17. 696
  18. 2
  19. 2216
  20. 6
  21. 8
  22. 3
  23. 4
  24. 5
  25. 1
  26. 3
  27. 4
  28. 3636
  29. 4
  30. 2
  31. 9
  32. 2622
  33. 5313
  34. 3
  35. 4
  36. 9
  37. 1
  38. 2
  39. 1308
  40. 2

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