Letpbe an odd prime of the formp= 4k+ 3

a) Prove that ifq= 2p+ 1 is prime, thenqdivides the Mersenne number $\displaystyle M_{p} = 2^p - 1$.

b) Use part (a) to prove that $\displaystyle M_{11}$, $\displaystyle M_{23}$, and $\displaystyle M_{251}$ are composite numbers.