have you considered the formulas for \(\displaystyle \varphi (n)\) for some integer \(\displaystyle n\)? the idea for the proofs should be evident after that

Show first that for m,n relatively prime \(\displaystyle \varphi(mn) = \varphi(m)\varphi(n)\)

Now, notice \(\displaystyle \frac{m}{\textrm{gcd}(m,n)}\) and \(\displaystyle n\) are relatively prime for any m,n and use the first result and the relation between gcd and lcm.

Yes, the division is well defined by the question 1) and you can use the fact that \(\displaystyle \frac{m}{\mathrm{gcd}(m,n)}\) and \(\displaystyle gcd(m,n)\) are relatively prime and distribute \(\displaystyle \varphi\)