Example 5 Calculate $\left[\frac{23 \times 1}{101}\right]+\left[\frac{23 \times 2}{101}\right]+\cdots+$ $\left[\frac{23 \times 100}{101}\right]$.

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Since 23 and 101 are coprime, for $k=1$, $2, \cdots, 100$, $\frac{23 k}{101}$ is not an integer. Therefore,
$$
\begin{array}{l}
\frac{23 k}{101}-1<\left[\frac{23 k}{101}\right]<\frac{23 k}{101}, \\
\frac{23(101-k)}{101}-1<\left[\frac{23(101-k)}{101}\right] \\
<\frac{23(101-k)}{101}. \\
\text{ Hence }\left(\frac{23 k}{101}-1\right)+\left[\frac{23(101-k)}{101}-1\right] \\
<\left[\frac{23 k}{101}\right]+\left[\frac{23(101-k)}{101}\right] \\
<\frac{23 k}{101}+\frac{23(101-k)}{101}, \\
21<\left[\frac{23 k}{101}\right]+\left[\frac{23(101-k)}{101}\right]<23. \\
\end{array}
$$

Therefore, $\left[\frac{23 k}{101}\right]+\left[\frac{23(101-k)}{101}\right]=22$.
Thus, we can pair $\left[\frac{23 \times 1}{101}\right],\left[\frac{23 \times 2}{101}\right], \cdots$, $\left[\frac{23 \times 100}{101}\right]$ from the beginning to the end, forming 50 pairs, with each pair summing to 22.
$$
\begin{array}{l}
\text{ Hence }\left[\frac{23 \times 1}{101}\right]+\left[\frac{23 \times 2}{101}\right]+\cdots+\left[\frac{23 \times 100}{101}\right] \\
=22 \times 50=1100. \\
\end{array}
$$