We are tasked with finding the sum of all prime numbers $p$ for which there is no integer solution to the congruence:

$3(6x+1) \equiv 4 \pmod{p}.$

Step 1: Simplify the congruence

First, expand the expression on the left-hand side:

$3(6x + 1) = 18x + 3.$

Thus, the congruence becomes:

$18x + 3 \equiv 4 \pmod{p}.$

Subtract 3 from both sides:

$18x \equiv 1 \pmod{p}.$

Step 2: Solve for $x$

We need to determine whether there exists an integer solution to the congruence:

$18x \equiv 1 \pmod{p}.$

This is equivalent to finding whether 18 has a multiplicative inverse modulo $p$, which occurs if and only if $18$ and $p$ are coprime, i.e., $\gcd(18, p) = 1$.

Step 3: Condition for no solution

If $\gcd(18, p) > 1$, then $18$ does not have an inverse modulo $p$, and hence there is no solution to the congruence $18x \equiv 1 \pmod{p}$. Therefore, we need to find primes $p$ for which $\gcd(18, p) > 1$.

Step 4: Analyze the prime factors of 18

The prime factorization of 18 is:

$18 = 2 \times 3^2.$

Thus, $\gcd(18, p) > 1$ if $p$ is either 2 or 3.

Step 5: Sum the relevant primes

The primes for which $\gcd(18, p) > 1$ are $p = 2$ and $p = 3$.

The sum of these primes is:

$2 + 3 = 5.$

Final Answer:

The sum of all prime numbers $p$ for which there is no integer solution to the congruence is $\boxed{5}$.

---

Would you like more details on any of the steps, or do you have additional questions?

Related questions:

1. How do you find the multiplicative inverse of a number modulo $p$?
2. What does it mean for two numbers to be coprime?
3. How do you solve a linear congruence modulo $p$?
4. Why does $\gcd(18, p) = 1$ guarantee an inverse exists?
5. How does the prime factorization of a number affect its congruences?

Tip:

When solving congruences, always check if the coefficient and the modulus are coprime—this determines whether an inverse exists.