[math-fun] Best proof that RSA Public-Key Cryptography works ?

I've had cause to revisit my notes on RSA Public Key Cryptography. The proof that it works is trivial if the message 'm' is coprime to the 'n' = p*q which features in the Public Key. I've seen no mentions of the case when m is divisible by p or q. Is there a neat proof handy that it doesn't matter if 'm' is divisible by p or q? Apologies as this is probably a noddy one, but thanks in advance. Guy