A nice Property of Euler function, prove via multi set.

We know that Euler function

$$\begin{array}{}\text{(2)}& \phi (n)=n\prod _{p|n}((1-\frac{1}{p})\end{array}$$

Proposition.

Let $gcd(s,t)=g$$\gcd(s,t)=g$

$$\begin{array}{}\text{(3)}& \phi (g)\phi (st)=g\phi (s)\phi (t)\end{array}$$

Proof.

The left hand side is

$$\begin{array}{}\text{(4)}& gst\prod _{p|g}(1-\frac{1}{p})\prod _{p^{\prime }|st}(1-\frac{1}{p^{\prime }})\end{array}$$

and the right hand side is

$$\begin{array}{}\text{(5)}& gst\prod _{p|s}(1-\frac{1}{p})\prod _{p^{\prime }|t^{\prime }}(1-\frac{1}{p^{\prime }})\end{array}$$

So we only need to prove that

$$\begin{array}{}\text{(6)}& t\downarrow \coprod s\downarrow =st\downarrow \coprod gcd(s,t)\end{array}$$

Which is extremely obvious. You doubt count the gcd part in the left hand side.

The notation is from coproduct, but if it is coproduct, we only can talk about isomorphism. So what I mean here is union of multi set.