Problem: Let $P_1^{}$ be a regular $r~\mbox{gon}$ and $P_2^{}$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each 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 $s_{}^{}$? Solution: We can organize the information above first: $ \begin{align*} & 180-\frac{360}{r}& =\frac{59}{58}\left(180-\frac{360}{s}\right) \\ \implies & \frac{59\cdot 360}{58s}-\frac{360}{r} & =\frac{1}{58} 180 \\ \implies & 59 \cdot 360 r-58 s \cdot 360 & =r s 180 \\ \implies & 59 \cdot 2 r-58\cdot 2 s & =r s \\ \implies & 118 r-1165& =r \\ \implies (s-118)(r+116) = -118 \cdot 116 \end{align*} $ From here, we can see that $s-118 < 0$ and the least $s$ that yields \integral $r$ is $s = 117$.