One, (20 points) Renovate a pedestrian path, changing the ground originally paved with $n^{2}-96$ identical square tiles to be paved with larger identical square tiles, requiring a total of $5 n+51$ tiles to complete the paving. If $n^{2}-96$ is divisible by $5 n+51$, find the value of the positive integer $n$.
Given the problem, let $\frac{n^{2}-96}{5 n+51}=k, k$ be a positive integer, then
$$
\begin{array}{l}
n^{2}-5 k n-(51 k+96)=0 . \\
\text { Therefore, } \Delta=25 k^{2}+4(51 k+96) \\
=25 k^{2}+204 k+384 .
\end{array}
$$

Since $n$ is an integer, $\Delta$ must be a perfect square.
$$
\text { And }(5 k+19)^{2}<25 k^{2}+204 k+384<(5 k+21)^{2} \text {, }
$$

Therefore, $\Delta=(5 k+20)^{2}=25 k^{2}+204 k+384$.
Solving for $k$ gives $k=4$.
Substituting $k=4$ into equation (1) yields
$$
n^{2}-20 n-300=0 \text {. }
$$

Solving for $n$ gives $n=30$.