Relative prime message in RSA encryption.

Question

Why has the message $P$ to be relative prime to $n$ in RSA encryption?

This should be fault? \begin{align} C &\equiv P^e \pmod{n} \\ &\equiv 101112^{11111357} \pmod{9998000099} \\ &\equiv 3316546434 \pmod{9998000099} \end{align}

Answer 1

$P$ can be any number $< n$, encryption will work. It will produce a valid ciphertext for which decryption still works.

It's also a very small probability (for realistically sized $n$) that this would happen anyway. If anyone were to randomly generate some $P$ and notice that the gcd of $P$ and $n$ was not $1$, that person would have factored $n$ and broken this RSA-instance.

Your example does have gcd of $P$ and $n$ equal to $1$.