II. (16 points) A plot of land can be covered by $n$ identical square tiles. If smaller identical square tiles are used, then $n+76$ such tiles are needed to cover the plot. It is known that $n$ and the side lengths of the tiles are integers. Find $n$.


Let the side length of the larger tile be $x$, and the side length of the smaller tile be $r$. Then $x, y$ are both natural numbers, and
$$
n x^{2}=(n+76) y^{2} \text {. }
$$

If $(x, y)=d$, let $x_{1}=\frac{x}{d}, y_{1}=\frac{y}{d}$, then $x_{1}, y_{1}$ are both natural numbers, and $\left(x_{1}, y_{1}\right)=1$. Thus, the previous equation can be transformed into
$$
n x_{1}^{2}=(n+76) y_{1}^{2} \text {. }
$$

We have $\frac{x_{1}^{2}}{y_{1}^{2}}=\frac{n+76}{n}, \frac{x_{1}^{2}-y_{1}^{2}}{y_{1}^{2}}=\frac{76}{n}$.
Since $\left(x_{1}^{2}-y_{1}^{2}, y_{1}^{2}\right)=1$, $\left(x_{1}-y_{1}\right)\left(x_{1}+y_{1}\right)$ is a divisor of 76 $=$ $2^{2} \times 19$. Noting that $x_{1}+y_{1}, x_{1}-y_{1}$ have the same parity, and $x_{1}+y_{1}>x_{1}-y_{1}$, we have
$$
\left\{\begin{array} { l } 
{ x _ { 1 } + y _ { 1 } = 1 9 , } \\
{ x _ { 1 } - y _ { 2 } = 1 }
\end{array} \text { or } \left\{\begin{array}{l}
x_{1}+y_{1}=2 \times 19, \\
x_{1}-y_{1}=2 .
\end{array}\right.\right.
$$

Since $\left(x_{1}, y_{1}\right)=1$, we can only have $\left\{\begin{array}{l}x_{1}=10 \\ y_{1}=9 .\end{array}\right.$
Thus, $\frac{76}{n}=\frac{19}{81}, n=324$.