# 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 $q \mathrel \backslash M_p$.

$q = 2 k p + 1$

and so by definition of congruence modulo an integer:

$q \equiv 1 \pmod p$

immediately.

$\blacksquare$