# 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.

## Problem

For some positive integer $k$, the repeating base- $k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? $\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$

## Solution 1

We can expand the fraction $0.\overline{23}_k$ as follows: $0.\overline{23}_k = 2\cdot k^{-1} + 3 \cdot k^{-2} + 2 \cdot k^{-3} + 3 \cdot k^{-4} + ...$ Notice that this is equivalent to $$2( k^{-1} + k^{-3} + k^{-5} + ... ) + 3 (k^{-2} + k^{-4} + k^{-6} + ... )$$

By summing the geometric series and simplifying, we have $\frac{2k+3}{k^2-1} = \frac{7}{51}$. Solving this quadratic equation (or simply testing the answer choices) yields the answer $k = \boxed{\textbf{(D) }16}$.

## Solution 2

Let $a = 0.2323\dots_k$. Therefore, $k^2a=23.2323\dots_k$.

From this, we see that $k^2a-a=23_k$, so $a = \frac{23_k}{k^2-1} = \frac{2k+3}{k^2-1} = \frac{7}{51}$.

Now, similar to in Solution 1, we can either test if $2k+3$ is a multiple of 7 with the answer choices, or actually solve the quadratic, so that the answer is $\boxed{\textbf{(D) }16}$.

## Solution 3 (bash)

We can simply plug in all the answer choices as values of $k$, and see which one works. After lengthy calculations, this eventually gives us $\boxed{\textbf{(D) }16}$ as the answer.

## Solution 4

Just as in Solution 1, we arrive at the equation $\frac{2k+3}{k^2-1}=\frac{7}{51}$.

We can now rewrite this as $\frac{2k+3}{(k-1)(k+1)}=\frac{7}{51}=\frac{7}{3\cdot 17}$. Notice that $2k+3=2(k+1)+1=2(k-1)+5$. As $17$ is a prime, we therefore must have that one of $k-1$ and $k+1$ is divisible by $17$. Now, checking each of the answer choices, this gives $\boxed{\textbf{(D) }16}$.

## Solution 5

Assuming you are familiar with the rules for basic repeating decimals, $0.232323..._(10) = \frac{23}{99}$. Now we want our base, $k$, to conform to $23 = 7\, (mod\, k)$ and $99 = 51\, (mod\, k)$, the reason being that we wish to convert the number from base $10$ to base $k$. Given the first equation, we know that $k$ must equal 9, 16, 23, or generally, $7n+2$. The only number in this set that is one of the multiple choices is $16$. When we test this on the second equation, $99 = 51\, (mod\, k)$, it comes to be true. Therefore, our answer is $\boxed{\textbf{(D) }16}$.

## Video Solution

For those who want a video solution: https://www.youtube.com/watch?v=DFfRJolhwN0

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 