Euler's totient function

Inclusion-exclusion. The number of integers in $\{1,\ldots,n\}$ coprime to $n$ equals $n$ minus those divisible by some prime $p\mid n$. By inclusion-exclusion:
$$\varphi(n)=n-\sum_{p\mid n}\frac np+\sum_{p<q\mid n}\frac{n}{pq}-\cdots=n\prod_{p\mid n}\left(1-\frac1p\right).$$

Conclusion. The product formula holds.

Example. $\varphi(12)=12(1-1/2)(1-1/3)=12\cdot\frac12\cdot\frac23=4$ (the integers $1,5,7,11$).