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
[hide]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 geometric series and simplifying, we have . Solving this quadratic equation (or simply testing the answer choices) yields the answer
.
Solution 2
Let . Therefore,
.
From this, we see that , so
.
Now, similar to in Solution 1, we can either test if is a multiple of 7 with the answer choices, or actually solve the quadratic, so that the answer is
.
Solution 3 (bash)
We can simply plug in all the answer choices as values of , and see which one works. After lengthy calculations, this eventually gives us
as the answer.
Solution 4
Just as in Solution 1, we arrive at the equation .
We can now rewrite this as . Notice that
. As
is a prime, we therefore must have that one of
and
is divisible by
. Now, checking each of 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 |
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