I'm quite stuck on this question and would really appreciate anyone who's able to explain the way it's worked out to me!
The full question is 'Let b be an integer coprime to 91. Assume that b is a quadratic residue modulo 91. Prove that 91 is pseudoprime to base b'.
Any help would be much appreciated! I've been staring at this question for a while now.
If $b\equiv x^2\bmod91$, then $b\equiv x^2\bmod 13$ and $7$.
Therefore, $b^6\equiv x^{12}\equiv1\bmod13$ and $7$, assuming $b$ is coprime to $13$ and $7$.
Therefore, $b^6\equiv1\bmod 91$.
Therefore $b^{90}\equiv 1\bmod 91$; i.e., $91$ is a pseudoprime to base $b$.