Example 4 (Euler's $\varphi$ function calculation formula)
Euler's $\varphi$ function value at $n$, $\varphi(n)$, is defined as the number of natural numbers that are coprime to $n$ in the set $\{1,2, \cdots, n\}$. Suppose $n$ has the standard factorization
$$n=p_{1}^{z_{1}} \cdots p_{s}^{\alpha_{s}},$$

where $p_{1}, \cdots, p_{s}$ are distinct prime numbers, $\alpha_{j} \geqslant 1,1 \leqslant j \leqslant s, s \geqslant 1$.
Let $P_{i}$ represent the property that a natural number in the set $S=\{1,2, \cdots, n\}$ is divisible by $P_{i}$ $(i=1, \cdots s)$. The subset of $S$ with property $P_{i}$ is denoted as $A_{i}$. Thus, we have
$$\begin{aligned}
\varphi(n)= & \left|\bar{A}_{1} \cap \cdots \cap \bar{A}_{s}\right|=|S|-\sum_{i}\left|A_{i}\right|+\sum_{i, j}\left|A_{i} \cap A_{j}\right|-+\cdots \\
& +(-1)^{s}\left|A_{1} \cap \cdots \cap A_{s}\right| \\
= & n-\sum_{i} \frac{n}{p_{i}}+\sum_{i<j} \frac{n}{p_{i} p_{j}}-+\cdots+(-1)^{s} \frac{n}{p_{1} \cdots p_{s}} \\
= & n\left(1-\frac{1}{p_{1}}\right) \cdots\left(1-\frac{1}{p_{s}}\right)=n \prod_{p \mid n}\left(1-\frac{1}{p}\right)
\end{aligned}$$

Example: From $60=2^{2} \cdot 3 \cdot 5$, we get
$$\varphi(60)=60\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)=16 .$$
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