Prove the following statement: If $n, m\in\mathbb{Z} $ and $g=$gcd$(n, m) $ then is $$\varphi(m, n) =\frac{ \varphi(m) \varphi(n) g} {\varphi(g)}. $$
Hint: Prove the statement with induction above the number of common prime divisors.
How can I do the first step of induction?
