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 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
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 .
Solution 4
Assuming you are familiar with the rules for basic repeating decimals, . Now we want our base, , to conform to and , the reason being that we wish to convert the number from base to base . Given the first equation, we know that must equal 9, 16, 23, or generally, . The only number in this set that is one of the multiple choices is . When we test this on the second equation, , it comes to be true. Therefore, our answer is .
Solution 5
Note that the LHS equals from which we see our equation becomes
Note that therefore divides but as is prime this therefore implies (Warning: This would not be necessarily true if were composite.) Note that is the only answer choice congruent satisfying this modular congruence, thus completing the problem.
~ Professor-Mom
Video Solution
For those who want a video solution: https://www.youtube.com/watch?v=DFfRJolhwN0
Video Solution
Education, the Study of Everything
(Please put video solutions at the end in order of when they were edited in)
Video Solution by TheBeautyofMath
https://youtu.be/FFpBMkKnOk8?t=1263
~IceMatrix
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.