Example 4 Find the number of integers between 1 and 250 that are divisible by any of the integers 2, 3, 5, and 7.

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Let $A$ denote the set of integers between 1 and 250 that are divisible by 2, $B$ the set of integers divisible by 3, $C$ the set of integers divisible by 5, and $D$ the set of integers divisible by 7. Thus,
$$
\begin{array}{l}
|A|=\left[\frac{250}{2}\right]=125 ;|B|=\left[\frac{250}{3}\right]=83 ; \\
|C|=\left[\frac{250}{5}\right]=50 ;|D|=\left[\frac{250}{7}\right]=35 ; \\
|A \cap B|=\left[\frac{250}{2 \times 3}\right]=41 ;|A \cap C|=\left[\frac{250}{2 \times 5}\right]=25 ; \\
|A \cap D|=\left[\frac{250}{2 \times 7}\right]=17 ;|B \cap C|=\left[\frac{250}{3 \times 5}\right]=16 ; \\
|B \cap D|=\left[\frac{250}{3 \times 7}\right]=11 ;|C \cap D|=\left[\frac{250}{5 \times 7}\right]=7 ; \\
|A \cap B \cap C|=\left[\frac{250}{2 \times 3 \times 5}\right]=8 ;|A \cap B \cap D|=\left[\frac{250}{2 \times 3 \times 7}\right]=5 ; \\
|A \cap C \cap D|=\left[\frac{250}{2 \times 5 \times 7}\right]=3 ;|B \cap C \cap D|=\left[\frac{250}{3 \times 5 \times 7}\right]=2 ; \\
|A \cap B \cap C \cap D|=\left[\frac{250}{2 \times 3 \times 5 \times 7}\right]=1 .
\end{array}
$$

Here, $[x]$ denotes the greatest integer not exceeding $x$. Therefore, $|A \cup B \cup C \cup D|=|A|+|B|+$
$$
\begin{array}{l}
|C|+|D|-|A \cap B|-|A \cap C|-|A \cap D|-|B \cap C|-|B \cap D|- \\
|C \cap D|+|A \cap B \cap C|+|A \cap B \cap D|+|A \cap C \cap D|+|B \cap C \cap D|- \\
|A \cap B \cap C \cap D|=193 .
\end{array}
$$