Example 3 The three sides of a triangle are integers $l, m, n$, 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 $x$. Find the minimum perimeter of such a triangle.
(2003 National High School Mathematics Competition)
From the given, we know that $3^{l} \equiv 3^{m} \equiv 3^{n}\left(\bmod 10^{4}\right)$, and then use Euler's theorem.
Solution From the given, we know: $3^{\prime} \equiv 3^{\prime m} \equiv 3^{n}\left(\bmod 10^{1}\right)$, by $3^{m} \equiv 3^{n}\left(\bmod 10^{4}\right) \Rightarrow 3^{n}\left(3^{m-n}-1\right) \equiv$ $0\left(\bmod 10^{4}\right)$, and $\left(3,10^{4}\right)=1$, so $3^{m-n}-1 \equiv 0\left(\bmod 10^{4}\right)$, i.e., $3^{m-n} \equiv 1\left(\bmod 10^{4}\right)$. Next, find the smallest $t \in \mathbf{N}^{\cdot}$ such that $3^{\prime} \equiv 1\left(\bmod 10^{4}\right) \Leftrightarrow 3^{\prime} \equiv 1\left(\bmod 2^{1}\right)$ and $3^{\prime} \equiv 1\left(\bmod 5^{\prime}\right)$. It is easy to verify that the smallest positive integer 4 makes $3^{4} \equiv 1\left(\bmod 2^{1}\right)$. Also, by Euler's theorem: $3^{43^{1}} \equiv 1\left(\bmod 5^{1}\right)$. That is, $3^{501} \equiv 1\left(\bmod 5^{1}\right)$. After verification, $3^{100} \not\equiv 1\left(\bmod 5^{4}\right)$, $3^{253} \not\equiv 1\left(\bmod 5^{4}\right)$, so the smallest positive integer here is 500, satisfying $3^{500} \equiv 1\left(\bmod 5^{4}\right)$. Therefore, the smallest $t$ that satisfies the condition is $[4,500]=500$, so $m-n=500 k_{1}, l-m=500 k_{2}$. Also, since $l, m, n$ are the sides of a triangle, we have: $m+n>l \Rightarrow n>500\left(k_{2}-k_{1}\right) \geqslant 500$. Taking $n=501, m=1001, l=1501$, the minimum perimeter of such a triangle is 3003.
/Comment Euler's theorem is an important tool for handling congruence problems.