Factor of Mersenne Number Mp equivalent to 1 mod p

Theorem
Let $p$ and $q$ be prime numbers such that $q$ is a divisor of the Mersenne number $M_p$.

Then:
 * $q \equiv 1 \pmod p$

Proof
Let $q \divides M_p$.

From Factor of Mersenne Number $M_p$ is of form $2 k p + 1$:
 * $q = 2 k p + 1$

and so by definition of congruence modulo an integer:
 * $q \equiv 1 \pmod p$

immediately.