x and y are positive integers with `x gt y` . The numbers `(x + 2y)` and `(2x+y)` leave remainders 6 and 1 respectively when divideb by 7. What is the remainder when (x -y) is divided by 7?

Options

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

Answer

To solve the problem, we need to find the remainder when \( x - y \) is divided by 7, given the conditions about \( x + 2y \) and \( 2x + y \).

### Step-by-Step Solution:

1. **Set Up the Equations**:
We know that:
- \( x + 2y \equiv 6 \mod 7 \)
- \( 2x + y \equiv 1 \mod 7 \)

This means:
\[ x + 2y = 7m + 6 \quad \text{(1)} \]
\[ 2x + y = 7n + 1 \quad \text{(2)} \]
where \( m \) and \( n \) are integers.

2. **Subtract Equation (1) from Equation (2)**:
We can subtract equation (1) from equation (2):
\[ (2x + y) - (x + 2y) = (7n + 1) - (7m + 6) \]
Simplifying the left side:
\[ 2x + y - x - 2y = x - y \]
Simplifying the right side:
\[ 7n + 1 - 7m - 6 = 7(n - m) - 5 \]
Thus, we have:
\[ x - y = 7(n - m) - 5 \quad \text{(3)} \]

3. **Determine the Remainder**:
From equation (3), we can see that when \( x - y \) is divided by 7, the remainder is \( -5 \). However, since remainders must be non-negative, we convert \( -5 \) into a positive remainder:
\[ -5 \equiv 2 \mod 7 \]
This is because \( -5 + 7 = 2 \).

4. **Conclusion**:
Therefore, the remainder when \( x - y \) is divided by 7 is \( 2 \).

### Final Answer:
The remainder when \( x - y \) is divided by 7 is **2**.