It is an easy exercise to prove your statement for all Mersenne primes, not just the known ones, using that $2^6\equiv 1\pmod{9}$. Indeed, $p$ must be prime for $M_p=2^p-1$ to be prime. Hence either $p\equiv 1\pmod{6}$ in which case $M_p\equiv 2^1-1\equiv 1\pmod{9}$ and so $M_p\equiv 1\pmod{18}$, or $p\equiv 5\pmod{6}$ in which case $M_p\equiv 2^5-1\equiv 4\pmod{9}$ and so $M_p\equiv 13\pmod{18}$. I told you earlier that your questions are not of research level. Try MathStackExchange next time.

It is an easy exercise to prove your statement for all Mersenne primes, not just the known ones, using that $2^6\equiv 1\pmod{9}$. Indeed, $p$ must be prime for $M_p=2^p-1$ to be prime. Hence either $p\equiv 1\pmod{6}$ in which case $M_p\equiv 2^1-1\equiv 1\pmod{9}$ and so $M_p\equiv 1\pmod{18}$, or $p\equiv 5\pmod{6}$ in which case $M_p\equiv 2^5-1\equiv 4\pmod{9}$ and so $M_p\equiv 13\pmod{18}$. I told you earlier that your questions are not of research level. Try MathStackExchange next time.