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
-- OmicronGamma
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
-eric2020
Solution 3 (extremely rigorous)
Plug in the values of k and bash. This gives us .
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.