2019 AMC 10A Problems/Problem 18
- The following problem is from both the 2019 AMC 10A #18 and 2019 AMC 12A #11, so both problems redirect to this page.
Contents
Problem
For some positive integer , the repeating base- representation of the (base-ten) fraction is . What is ?
Solution 1
We can expand the fraction as follows: . Notice that this is equivalent to
By summing the infinite series and simplifying, we have . Solving this quadratic equation or testing the answer choices yields the answer
Solution 2
Let . Therefore, .
From this, we see that .
Solving for a:
Similar to Solution 1, testing if is a multiple of 7 with the answer choices or solving the quadratic yields , so the answer is .
Solution 3 (extremely rigorous)
Plug in the values of k and bash. This gives us .
Solution 4
Similar to Solution 1, we arrive at . We can rewrite this as . Notice that . As is a prime, we have that one of and is divisible by . Looking at the answer choices, this gives .
Video Solution
For those who want a video solution : https://www.youtube.com/watch?v=DFfRJolhwN0
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.