Multiplicativity of $\varphi$

CRT. The isomorphism $x\mapsto(x\bmod m,x\bmod n)$ sends units of $\mathbb{Z}/mn\mathbb{Z}$ to pairs of units.

Count. The number of pairs $(a,b)$ with $a$ a unit mod $m$ and $b$ a unit mod $n$ is $\varphi(m)\varphi(n)$.

Conclusion. $\varphi(mn)=\varphi(m)\varphi(n)$ when $\gcd(m,n)=1$.

Corollary. $\varphi(n)=n\prod_{p\mid n}(1-1/p)$.