What’s the proof that the Euler totient function is multiplicative?

That is,

why is $$\varphi(AB)=\varphi(A)\varphi(B)$$, if $$A$$ and $$B$$ are two coprime positive integers?

It’s not just a technical trouble—I can’t see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most.
And why the minimum value for $$\varphi (n)$$ should be $$\frac{4n}{15}$$ completely passes me by.

In general, if $R$ and $S$ are rings, then $R\times S$ is a ring. The units of $R\times S$ are the elements $(r,s)$ with $r$ a unit of $R$ and $s$ a unit of $S$. If $R$ and $S$ are finite rings, the number of units in $R\times S$ is therefore the number of units in $R$ times the number of units in $S$.
Now if $\gcd(A,B)=1$, then $$\mathbb Z/\left<AB\right> \cong \mathbb Z/\left<A\right> \times \mathbb Z/\left<B\right>$$
But the number of units in the ring $\mathbb Z/\left<n\right>$ is $\phi(n)$. So the number of units in $\mathbb Z/\left<AB\right>$ is $\phi(AB)$ and the number of units in $\mathbb Z/\left<A\right> \times \mathbb Z/\left<B\right>$ is $\phi(A)\phi(B)$