Prove: If $ p|2^q -1$, p,q primes numbers then p is congruent to 1 modulo q

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p|2^q -1, so 2^q - 1 = pk and 2^q = pk - 1

from Fermat we've got 2^q is congruent to 2 (mod q) pk-1 is congruent to 2 (mod q) pk is congruent to 1 (mod q), then k must be 1

Is this evidence is correct ?