TITLE: How to prove $\phi(mn) > \phi(m)\phi(n)$ if $(m,n) \ne 1$
QUESTION [9 upvotes]: I need to prove that 
$$\phi(mn) > \phi(m)\phi(n)$$
if $m$ and $n$ have a common factor greater than 1. 
I have read up on the case where $m$ and $n$ are relatively prime, then $\phi(mn)=\phi(m)\phi(n)$.

REPLY [8 votes]: According this recently asked question, 
$$
  \phi(mn) = \phi(m) \phi(n) \frac{d}{\phi(d)} ,
$$
where $d = \gcd(m,n)$.
Your question follows from the fact that $\varphi(d) < d$ whenever $d>1$.