<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 $p$.
</Question>

<think>
<Plan_1>
Okay, let's try to figure out this problem. So, the question is asking for the sum of all finite primes p where the congruence 8x ≡ 1 mod p has no solutions. Hmm, right. Alright, first, I need to recall when a linear congruence like ax ≡ b mod p has a solution.
I remember that such an equation has a solution if and only if the greatest common divisor (gcd) of a and p divides b. Since p is a prime here, the gcd of 8 and p will be either 1 or 8, right? Because primes are only divisible by 1 and themselves. So, if p is not 2, then gcd(8, p) is 1, since 8 is 2^3 and if p is not 2, they share no common factors other than 1.
But if p is 2, then gcd(8, 2) is 2. In that case, for the congruence 8x ≡ 1 mod 2 to have a solution, the gcd (which is 2) must divide 1. But 2 doesn't divide 1, so there's no solution when p = 2. Wait, but hold on, maybe I need to check that again.

Let me think. The congruence ax ≡ b mod p has a solution if and only if gcd(a, p) divides b. Here, a is 8, b is 1, and p is prime. So, if p ≠ 2, then gcd(8, p) = 1, which divides 1, so there is a solution. But if p = 2, then gcd(8, 2) = 2, and 2 does not divide 1, so there is no solution.
Therefore, the only prime p for which 8x ≡ 1 mod p has no solution is p = 2. Hence, the sum of all such primes is 2.

Wait, but that seems too straightforward. Let me verify again. Maybe there's another prime where 8 and p are not coprime, but p divides 1?

<prompt_1.1> Re-evaluate the condition for the solvability of the congruence 8x ≡ 1 mod p and confirm the result. </prompt_1.1>

But let me check with an example. Let's take p = 3.

<prompt_1.2> Test the congruence 8x ≡ 1 mod p for small primes to confirm the result. </prompt_1.2>

Wait, but let me check another prime, say p = 7.

<prompt_1.3> Re-evaluate the congruence 8x ≡ 1 mod p for additional small primes to confirm the result. </prompt_1.3>

But wait, maybe I made a mistake here. Let me think again. The condition is that gcd(8, p) divides 1. Since gcd(8, p) is either 1 or 2 (if p = 2), then for primes other than 2, gcd is 1, which divides 1, so solution exists.

<prompt_1.4> Perform a final verification of the solution and conclude the sum of primes. </prompt_1.4>
</Plan_1>

<Execution_1>
<execution_1.1>
Wait, but if p divides 8, then 8 ≡ 0 mod p, so 0*x ≡ 1 mod p, which is impossible. So, if 8 ≡ 0 mod p, then the equation becomes 0 ≡ 1 mod p, which is impossible. So, when does 8 ≡ 0 mod p? That's when p divides 8. Since p is prime, the only possibility is p = 2. Because 8's prime factorization is 2^3. So, p = 2 is the only prime where 8 ≡ 0 mod p. Therefore, for p = 2, the equation 8x ≡ 1 mod 2 becomes 0 ≡ 1 mod 2, which is impossible. Hence, no solution.

Therefore, the only prime p where the congruence has no solution is p = 2. Therefore, the sum of all such primes is 2.
</execution_1.1>

<execution_1.2>
Then 8x ≡ 1 mod 3. Since 8 mod 3 is 2, so this reduces to 2x ≡ 1 mod 3. The inverse of 2 mod 3 is 2, because 2*2 = 4 ≡ 1 mod 3. So x = 2 is a solution. So, p = 3 is okay. How about p = 5? 8 mod 5 is 3, so 3x ≡ 1 mod 5. The inverse of 3 mod 5 is 2, since 3*2 = 6 ≡ 1 mod 5. So x = 2. So, p = 5 is okay. p = 7, as above, works. How about p = 17? 8 and 17 are coprime, so inverse exists.
</execution_1.2>

<execution_1.3>
If p = 7, then 8 ≡ 1 mod 7, so the equation becomes x ≡ 1 mod 7, which obviously has a solution x = 1. For p = 5, 8 ≡ 3 mod 5, so we need to solve 3x ≡ 1 mod 5. The inverse of 3 mod 5 is 2 because 3*2 = 6 ≡ 1 mod 5. So, x = 2 is a solution. For p = 17, 8 and 17 are coprime, so inverse exists. Let's see, 8*15 = 120 ≡ 1 mod 17 because 17*7=119, 120-119=1. So x=15 works. 

So, seems like for primes other than 2, 8 has an inverse modulo p. The only problem is when p divides 8, but since p is prime, the only such prime is 2. So, indeed, p=2 is the only prime where 8x ≡1 mod p has no solution.
</execution_1.3>

<execution_1.4>
For p = 2, gcd(8, 2) = 2, which does not divide 1, so no solution. Therefore, only p = 2 is excluded. Hence, the sum is 2.

But wait, let me check with another prime. Let's take p = 17. Then, 8x ≡ 1 mod 17. The inverse of 8 mod 17 is 15 because 8*15 = 120 ≡ 1 mod 17. So, there is a solution. Similarly, p = 5. 8x ≡ 1 mod 5. 8 mod 5 is 3, so 3x ≡ 1 mod 5. The inverse of 3 mod 5 is 2, since 3*2 = 6 ≡ 1 mod 5. So, x = 2. Therefore, solution exists. So, primes other than 2, there is a solution. So, only p = 2 is excluded. Therefore, the sum is 2.
</execution_1.4>
</Execution_1>

<Final_answer>
Therefore, the sum of all primes $p$ for which the congruence $8x\equiv 1\pmod{p}$ has no solutions $x$ is $\boxed{2}$.
</Final_answer>
</think>