To solve the congruence $3x \equiv 4 \pmod{7}$, we need to find the value of $x$ that satisfies this equation modulo 7.

Steps:

1. Simplify the congruence: $3x \equiv 4 \pmod{7}$ The goal is to isolate $x$. First, we need to find the multiplicative inverse of 3 modulo 7, because multiplying both sides of the congruence by this inverse will help us solve for $x$.

2. Find the multiplicative inverse of 3 modulo 7: The multiplicative inverse of 3 modulo 7 is the number $y$ such that: $3y \equiv 1 \pmod{7}$ We can test values of $y$ to find this:

   • For $y = 1$, $3 \times 1 = 3 \not\equiv 1 \pmod{7}$
   • For $y = 2$, $3 \times 2 = 6 \not\equiv 1 \pmod{7}$
   • For $y = 3$, $3 \times 3 = 9 \equiv 2 \pmod{7}$
   • For $y = 4$, $3 \times 4 = 12 \equiv 5 \pmod{7}$
   • For $y = 5$, $3 \times 5 = 15 \equiv 1 \pmod{7}$

   Therefore, the inverse of 3 modulo 7 is 5.

3. Multiply both sides of the original congruence by 5: Now, multiply both sides of $3x \equiv 4 \pmod{7}$ by 5: $5 \times 3x \equiv 5 \times 4 \pmod{7}$ Simplifying: $15x \equiv 20 \pmod{7}$ Since $15 \equiv 1 \pmod{7}$ and $20 \equiv 6 \pmod{7}$, this reduces to: $x \equiv 6 \pmod{7}$

Thus, the solution to the congruence $3x \equiv 4 \pmod{7}$ is: $x \equiv 6 \pmod{7}$

Summary:

The value of $x$ that satisfies $3x \equiv 4 \pmod{7}$ is $x = 6$, modulo 7.

Would you like further clarification on this, or have any additional questions?

Related questions:

1. How can I find the multiplicative inverse of any number modulo $n$?
2. What are the properties of modular arithmetic that are useful in solving congruences?
3. How can you solve a system of congruences using the Chinese Remainder Theorem?
4. How can I use modular arithmetic in cryptography algorithms like RSA?
5. What is the general approach to solving linear congruences?

Tip: When trying to solve congruences, always first look for a way to simplify the equation, like finding inverses or reducing both sides modulo $n$.