Factor of Mersenne Number Mp is of form 2kp + 1

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

Then:
 * $q = 2 k p + 1$

for some integer $k$.

Proof
Let $q \divides M_p$.

Then:
 * $2^p \equiv 1 \pmod q$

From Integer to Power of Multiple of Order, the multiplicative order of $2 \pmod q$ divides $p$.

By Fermat's Little Theorem, the multiplicative order of $2 \pmod q$ also divides $q - 1$.

Hence:
 * $q - 1 = 2 k p$