Let $P_1^{}$ be a [regular](https://artofproblemsolving.com/wiki/index.php/Regular_polygon) $r~\mbox{gon}$ and $P_2^{}$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each [interior angle](https://artofproblemsolving.com/wiki/index.php/Interior_angle) of $P_1^{}$ is $\frac{59}{58}$ as large as each interior angle of $P_2^{}$. What's the largest possible value of $s_{}^{}$?
The formula for the interior angle of a regular sided [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) is $\frac{(n-2)180}{n}$. 
Thus, $\frac{\frac{(r-2)180}{r}}{\frac{(s-2)180}{s}} = \frac{59}{58}$. Cross multiplying and simplifying, we get $\frac{58(r-2)}{r} = \frac{59(s-2)}{s}$. Cross multiply and combine like terms again to yield $58rs - 58 \cdot 2s = 59rs - 59 \cdot 2r \Longrightarrow 118r - 116s = rs$. Solving for $r$, we get $r = \frac{116s}{118 - s}$.
$r \ge 0$ and $s \ge 0$, making the [numerator](https://artofproblemsolving.com/wiki/index.php/Numerator) of the [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) positive. To make the [denominator](https://artofproblemsolving.com/wiki/index.php/Denominator) [positive](https://artofproblemsolving.com/wiki/index.php/Positive), $s < 118$; the largest possible value of $s$ is $117$.
This is achievable because the denominator is $1$, making $r$ a positive number $116 \cdot 117$ and $s = \boxed{117}$.