solution: 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 $\boxed{2}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  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 $p$.
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 2
extra_info.question: 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 $p$.
extra_info.split: test