Example 2 Find all natural numbers $n$ such that $13 \mid 7^n-1$.

 untranslated text:
例 2 求出所有使 $13 \mid 7^{\prime \prime}-1$ 的自然数 $n$.

 translated text:
Example 2 Find all natural numbers $n$ such that $13 \mid 7^n-1$.
Analysis Using Fermat's Little Theorem, we know that $13 \mid 7^{12}-1$, and then proceed further. $13 \mid 7^{12 k}-1$, while $13 \nmid 7^{12 k+r}-1,(1 \leqslant r<12)$. In fact, $7^{12 k+r}-1=7^{r}\left(7^{12 k}-1\right)+7^{r}-1$, so $13 \nmid 7^{12 k+r}-1,(1 \leqslant r<12)$. Therefore, the condition is satisfied if and only if $n=12 k\left(k \in \mathbf{N}^{*}\right)$. /Comment Fermat's Little Theorem is one of the important tools for handling such divisibility problems.