There are finitely many primes $p$ for which the congruence $$8x\equiv 1\pmod{p}$$has no solutions $x$. Determine the sum of all such $p$.

A solution exists if and only if 8 is invertible modulo p. In other words, \(\gcd(8,p)=1\). Since \(8=2^3\) is a power of 2, 8 is invertible modulo q if and only if q is an odd integer. All primes except for 2 are odd, so the number we are looking for is 2.