The given equation is:

\[
4x \equiv 10 \ (\text{mod} \ 14)
\]

We need to solve this congruence equation for \( x \). Let's break it down step by step:

### Step 1: Simplify the equation
Since this is a modulo equation, we're working in the system of integers modulo 14. The equation reads that \( 4x \) is congruent to 10 modulo 14, which means:
\[
4x - 10 \text{ is divisible by } 14.
\]

### Step 2: Check if a solution exists
We want to simplify \( 4x \equiv 10 \ (\text{mod} \ 14) \). First, note that 4 and 14 have a greatest common divisor (GCD) of 2. Therefore, the equation can only have a solution if the right-hand side, 10, is divisible by this GCD (2), which it is.

### Step 3: Divide by the GCD (which is 2)
We can now divide the entire equation by 2:

\[
2x \equiv 5 \ (\text{mod} \ 7)
\]

### Step 4: Solve the new congruence
Now, we need to solve:

\[
2x \equiv 5 \ (\text{mod} \ 7)
\]

We need to find the multiplicative inverse of 2 modulo 7. To find this, we test values of \( y \) such that:

\[
2y \equiv 1 \ (\text{mod} \ 7)
\]

Through trial or using the Extended Euclidean Algorithm, we find that \( y = 4 \), because:

\[
2 \times 4 = 8 \equiv 1 \ (\text{mod} \ 7)
\]

### Step 5: Multiply both sides by the inverse of 2 (which is 4)
Now, multiply both sides of \( 2x \equiv 5 \ (\text{mod} \ 7) \) by 4:

\[
x \equiv 5 \times 4 \ (\text{mod} \ 7)
\]

\[
x \equiv 20 \ (\text{mod} \ 7)
\]

Since \( 20 \equiv 6 \ (\text{mod} \ 7) \), we have:

\[
x \equiv 6 \ (\text{mod} \ 7)
\]

### Step 6: General solution
Now that we know \( x \equiv 6 \ (\text{mod} \ 7) \), we can write the general solution. Since we divided by 2 earlier, the original modulus was 14, so the general solution is:

\[
x = 6 + 7k \quad \text{for some integer } k.
\]

Thus, the solution to the equation \( 4x \equiv 10 \ (\text{mod} \ 14) \) is:

\[
x = 6 + 7k \quad \text{for any integer } k.
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do we find the multiplicative inverse in modular arithmetic?
2. What happens if the GCD is not a divisor of the right-hand side in a modular equation?
3. Can you explain how the Extended Euclidean Algorithm helps in finding the inverse?
4. How do you determine if a modular equation has no solution?
5. What are some real-world applications of modular arithmetic?

**Tip:** In modular arithmetic, when solving \( ax \equiv b \ (\text{mod} \ m) \), always check the GCD of \( a \) and \( m \) to see if a solution exists!

Learn how to solve the congruence equation 4x ≡ 10 (mod 14) step by step using modular arithmetic, GCD, and the multiplicative inverse. This solution explores the simplification process and the Extended Euclidean Algorithm. Ideal for Grades 9-12.

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