RSA and modular arithmetic

Coprime case. If $\gcd(m,n)=1$: by Euler, $m^{\varphi(n)}\equiv1\pmod n$. Since $ed\equiv1\pmod{\varphi(n)}$, $ed=1+k\varphi(n)$, so $m^{ed}=m\cdot(m^{\varphi(n)})^k\equiv m\pmod n$.

Non-coprime case. If $\gcd(m,n)>1$, then WLOG $p\mid m$. Then $m^{ed}\equiv0\pmod p$ and $m^{ed}\equiv m\pmod q$ (by Euler mod $q$). By CRT, $m^{ed}\equiv m\pmod n$ (and $m\equiv0\pmod p$ ensures $m^{ed}\equiv m\pmod p$ too).

Conclusion. RSA correctness holds for all $m$.