Example 7 Given that there are exactly 600 integer-sided triangles with unequal sides and the longest side exactly $n$. Find the value of $n$.
Explanation: The key to solving this problem is to establish an equation about $n$. Note that one side of the triangle, $n$, is fixed, and the three sides $x, y, n$ are all integers. Therefore, the number of such triangles is equal to the number of lattice points $(x, y)$ in the coordinate plane.

Let the lengths of the other two sides of the triangle be $x, y$. Thus, the number of positive integer solutions $(x, y)$ that satisfy the inequality system $\left\{\begin{array}{l}x+y>n, \\ x<y<n\end{array}\right.$ is the number of such triangles.

When $\dddot{n}$ is odd, the number of triangles is the number of lattice points inside $\triangle B C D$ in Figure 4, i.e.,
$$
\begin{aligned}
f(n) & =2+4+6+\cdots+(n-3) \\
& =\frac{1}{4}(n-1)(n-3) .
\end{aligned}
$$

When $n$ is even, the number of triangles is the number of lattice points inside $\triangle B C D$ in Figure 5, i.e.,
$$
\begin{array}{l} 
f(n)=1+3+5+\cdots+(n-3) \\
=\frac{1}{4}(n-2)^{2} . \\
\text { If } \frac{1}{4}(n-1)(n-3)=600, \text { then } n=51 ; \\
\text { If } \frac{1}{4}(n-2)^{2}=600 \text {, then there is no integer solution. }
\end{array}
$$

Therefore, the required $n=51$.
Note: This problem can also be solved by examining the length of the shortest side one by one, establishing an equation about $n$ and solving it.