Factor of Mersenne Number Mp equivalent to 1 mod p

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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 $q \mathrel \backslash M_p$.

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.

$\blacksquare$


Sources