Prove: When $m > 1$ and $n > 1$ and $\gcd(m, n) > 1$. Then $\phi(mn) \neq \phi(m)\phi(n)$.
Proof
Let $m,n \in \mathbb{Z}$ s.t. $\gcd(m,n) > 1$. Consider the congruences:
$x \equiv b \mod m$
$x \equiv c \mod n$
This is where I am stuck. If i were to prove the opposite of this statement I would appeal to the chinese remainder theorem and state that distinct solutions have a correspondence to distinct pairs. So I think is in the same, vein. We do not have a distinct correspondence, so somehow we will end up with: $\phi(mn) \neq \phi(m)\phi(n)$