AIME 2024b Problem 14

Let $b\ge 2$ be an integer. Call a positive integer $n$ $b\text-\textit{eautiful}$ if it has exactly two digits when expressed in base $b$ and these two digits sum to $\sqrt n$. For example, $81$ is $13\text-\textit{eautiful}$ because $81 = \underline{6} \ \underline{3}_{13}$ and $6 + 3 = \sqrt{81}$. Find the least integer $b\ge 2$ for which there are more than ten $b\text-\textit{eautiful}$ integers.