What is the sum of all prime numbers p for which there is no integer solution for x in the congruence $3(6x+1) \\equiv 4 \\pmod p$?




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.

What is the sum of all prime numbers p for which there is no integer solution for x in the congruence $3(6x+1) \equiv 4 \pmod p$?

Explore the solution to finding the sum of prime numbers p for which the congruence $3(6x+1) \equiv 4 \pmod p$ has no integer solution. This problem involves key concepts in modular arithmetic, gcd, and finding multiplicative inverses. The relevant primes are p = 2 and p = 3, and their sum is 5. This solution is suitable for students in Grades 10-12.

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