To solve the equation $5x \equiv 4 \mod 3$, let's break it down step by step:

Step 1: Simplify the equation modulo 3

We want to simplify both sides of the equation modulo 3.

• $5 \mod 3$ is equivalent to $2$, because when you divide 5 by 3, the remainder is 2.
• $4 \mod 3$ is equivalent to $1$, because when you divide 4 by 3, the remainder is 1.

So, the equation becomes: $2x \equiv 1 \mod 3$

Step 2: Solve for $x$

Now, we need to find the value of $x$ that satisfies $2x \equiv 1 \mod 3$.

We can test different values of $x \mod 3$ to find a solution:

• If $x = 0$, then $2(0) = 0 \mod 3$.
• If $x = 1$, then $2(1) = 2 \mod 3$.
• If $x = 2$, then $2(2) = 4 \mod 3 = 1 \mod 3$.

So, $x = 2$ satisfies the equation $2x \equiv 1 \mod 3$.

Final Answer:

$x \equiv 2 \mod 3$

Would you like to explore more about modular arithmetic or need further clarification? Here's a set of related questions to dive deeper into the topic:

1. How do we solve modular equations of higher degrees?
2. What are inverse elements in modular arithmetic and how do we find them?
3. How does the Chinese Remainder Theorem apply to systems of congruences?
4. Can you explain why modular arithmetic is used in cryptography?
5. What is the general method for solving linear congruences like this one?

Tip: When solving modular equations, always check if the coefficients have an inverse modulo the modulus. If they do, you can multiply both sides of the equation by that inverse to isolate $x$.