 Solve ( 2x \equiv 4 \ (\text{mod} \ 6) ).

To solve the congruence \( 2x \equiv 4 \ (\text{mod} \ 6) \), let's follow these steps:

### Step 1: Simplify the congruence
We can divide the entire congruence by 2, since the greatest common divisor (gcd) of 2 and 6 is 2 (so division by 2 is valid). This simplifies to:

\[
x \equiv 2 \ (\text{mod} \ 3)
\]

### Step 2: Solve the simplified congruence
The congruence \( x \equiv 2 \ (\text{mod} \ 3) \) means that \( x \) must satisfy:

\[
x = 3k + 2
\]

for some integer \( k \).

### Step 3: General solution
Now, since the original modulus was 6, we must find solutions modulo 6. The general solution is:

\[
x = 6n + 5
\]

where \( n \) is any integer.

Thus, the solution to the congruence \( 2x \equiv 4 \ (\text{mod} \ 6) \) is:

\[
x \equiv 5 \ (\text{mod} \ 6)
\]

This means that any integer of the form \( x = 6n + 5 \), for integer \( n \), is a solution.

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Do you want more details or have any questions?

Here are 5 related questions to explore this topic further:

1. How do you solve a congruence when the coefficient of \( x \) is not divisible by the modulus?
2. What is the meaning of the phrase "solving modulo \( n \)" in number theory?
3. How can you use the Chinese Remainder Theorem for solving systems of congruences?
4. What is the general method for solving linear Diophantine equations?
5. How does gcd (greatest common divisor) influence the solvability of congruences?

**Tip:** When simplifying a congruence, always check if you can divide both sides by a common factor of the coefficient and the modulus.

Explore how to solve the modular congruence 2x ≡ 4 (mod 6) step-by-step. Learn about number theory, congruences, and the use of the GCD in simplifying modular problems. This solution is ideal for students in Grades 9-12.

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