Example 3 For every positive integer $n$, how many primitive right triangles are there such that their area (numerically) equals $n$ times their perimeter?
Let the sides of a primitive right-angled triangle be $x, y, z$. Then $x, y$ are one odd and one even. Without loss of generality, assume $2 \mid y$. By the theorem, there exist $u, v \in \mathbf{N}^{*}, (u, v)=1$, and $u, v$ are one odd and one even, such that
$$x=u^{2}-v^{2}, y=2 u v, z=u^{2}+v^{2}.$$

By the given condition, we have $\frac{1}{2} x y=n(x+y+z)$, which means
$$u v\left(u^{2}-v^{2}\right)=n\left(2 u^{2}+2 u v\right),$$

Thus,
$$v(u-v)=2 n$$

Let the standard factorization of $n$ be $n=2^{r} p_{1}^{\varepsilon_{1}} p_{2}^{\sigma_{2}} \cdots p_{k}^{\sigma_{k}}$, where $p_{1}, p_{2}, \cdots, p_{k}$ are the odd prime factors of $n$.

Since $u, v$ are one odd and one even, $u-v$ is odd. From (4), we know $v$ is even. Additionally,
$$(u, v)=1,$$

So $(u-v, v)=1$. From (4), we can conclude
$$2^{r+1} \mid v$$

Let $v=2^{r+1} A, u-v=B$. Then from (4) and the above discussion, we know that for each $i=1,2, \cdots, k$, either $p_{i}^{s_{i}}\left|A, p_{i}\right| A$ or $p_{i}^{s_{i}^{*}}\left|B, p_{i}\right| B$ must hold, but not both. Therefore, there are exactly $2^{k}$ possible values for $v$ and $u-v$.

In conclusion, there are exactly $2^{k}$ primitive right-angled triangles that satisfy the condition, where $k$ is the number of odd prime factors of $n$.