Find x, if `9x -= 2` (mod 7).

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the equation \(9x \equiv 2 \mod 7\), we will follow these steps:

### Step 1: Simplify the equation
First, we need to simplify \(9x\) modulo \(7\). We can do this by finding the equivalent of \(9\) modulo \(7\):
\[ 9 \mod 7 = 2 \]
So, we can rewrite the equation:
\[ 2x \equiv 2 \mod 7 \]

### Step 2: Isolate \(x\)
Next, we want to isolate \(x\). To do this, we can divide both sides of the equation by \(2\). However, we need to ensure that \(2\) has a multiplicative inverse modulo \(7\). The inverse of \(2\) modulo \(7\) is \(4\) because:
\[ 2 \times 4 = 8 \equiv 1 \mod 7 \]
Now, we can multiply both sides of the equation \(2x \equiv 2\) by \(4\):
\[ 4 \cdot (2x) \equiv 4 \cdot 2 \mod 7 \]
This simplifies to:
\[ 8x \equiv 8 \mod 7 \]

### Step 3: Reduce modulo \(7\)
Now, we simplify \(8\) modulo \(7\):
\[ 8 \mod 7 = 1 \]
So, we have:
\[ x \equiv 1 \mod 7 \]

### Step 4: Conclusion
Thus, the solution to the equation \(9x \equiv 2 \mod 7\) is:
\[ x = 1 \]