Find all $m,n$ such that $\varphi(mn) = \varphi(n)$. [duplicate]

Characterize all pairs of integers $m$ and $n$ for which $\phi(mn) = \phi(n)$

I know that if $\gcd(m,n) = 1$, then $\phi(mn) = \phi(m)\phi(n)$

How can I use the fact to prove $\phi(mn) = \phi(n)$?

It easy to show that for all pairs $(m,n)$ the following identity holds: $$ \varphi(mn)=\varphi(m)\varphi(n)\frac{d}{\varphi(d)}, $$ where $d=\gcd(m,n)$.

So, it's enough to describe all pairs $(m,d)$, where $d\mid m$ and $\varphi(d)=d\varphi(m)$. Since $\varphi(d)\leqslant d$ and $1\leqslant \varphi(m)$ for all numbers $m$ and $d$, we conclude that $\varphi(m)=1$ and $\varphi(d)=d$.

Therefore, $m=1$ or $m=2$ and $d=1$. So, the answer is all pairs of the forms $(1,k)$ and $(2, 2k+1)$.