Example 5: There are $L$ squares arranged on a circle, one of which contains 1 chess piece. The chess piece moves counterclockwise around the circle: it moves 1 square the first time, 2 squares the second time, ... and $n$ squares the $n$-th time. Try to answer: which squares can the chess piece reach during its movement?

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Solution: Number the squares sequentially as $0,1,2,3,4$, and assume without loss of generality that the chess piece is in the square numbered 0. Then, the number of the square reached after the $n$th move is the remainder of $1+2+\cdots+n$ divided by 5.

For any positive integer $n$, let $n \equiv r(\bmod 5)(0 \leqslant r < 5)$. Then
$$
\begin{array}{l}
1+2+\cdots+n \\
=\frac{n(n+1)}{2}=\frac{r(r+1)}{2}(\bmod 5) .
\end{array}
$$

When $r=0,1,2,3,4$,
$$
\frac{r(r+1)}{2} \equiv 0,1,3,1,0(\bmod 5) \text {. }
$$

Thus, $\frac{n(n+1)}{2} \equiv 0,1,3(\bmod 5)$.
This indicates that the chess piece can never reach the squares numbered $2$ and $4$.

After the first move, it reaches square 1; after the second move, it reaches square 3; and after the fourth move, it returns to square 0. Therefore, the chess piece can reach the squares numbered $0,1,3$ during its movement.