Example 1 Suppose the lengths of the three sides of a triangle are integers $l$, $m$, and $n$, and $l > m > n$. It is known that
$$
\left\{\frac{3^{l}}{10^{4}}\right\}=\left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\},
$$

where $\{x\}=x-[x]$, and $[x]$ represents the greatest integer not exceeding the real number $x$. Find the minimum perimeter of such a triangle.
(2003, National High School Mathematics Competition)
Solution: Note that, when $a \equiv b(\bmod m)(0 \leqslant b < m)$, we get $500+2 n>1000+n$.
So, $n>500$.
Thus, $n \geqslant 501, m \geqslant 1001, l \geqslant 1501$.
Therefore, $m+n+l \geqslant 3003$, which means the minimum perimeter of the triangle is 3003.