If H.C.F. (56, 72)=`56x+72y`, then `x` and `y` are equal to:

Options

A. `x = 4, y=3`
B. `x = -4, y=3`
C. `x = 4, y=-3`
D. `x = -4, y=-3`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that the equation \( H.C.F.(56, 72) = 56x + 72y \) holds true.

### Step-by-Step Solution:

1. **Find the H.C.F. of 56 and 72**:
- First, we will find the prime factorization of both numbers.
- For 56:
\[ 56 = 2^3 \times 7 \]
- For 72:
\[ 72 = 2^3 \times 3^2 \]
- The common prime factors are \( 2^3 \). Therefore, the H.C.F. is:
\[ H.C.F.(56, 72) = 2^3 = 8 \]

2. **Set up the equation**:
- We have established that:
\[ 56x + 72y = 8 \]

3. **Substituting the options**:
- We will substitute the values of \( x \) and \( y \) from the given options to find which one satisfies the equation.

**Option 1**: \( x = 4, y = 3 \)
\[ 56(4) + 72(3) = 224 + 216 = 440 \quad \text{(not equal to 8)} \]

**Option 2**: \( x = -4, y = 3 \)
\[ 56(-4) + 72(3) = -224 + 216 = -8 \quad \text{(not equal to 8)} \]

**Option 3**: \( x = 4, y = -3 \)
\[ 56(4) + 72(-3) = 224 - 216 = 8 \quad \text{(equal to 8)} \]

**Option 4**: \( x = -4, y = -3 \)
\[ 56(-4) + 72(-3) = -224 - 216 = -440 \quad \text{(not equal to 8)} \]

4. **Conclusion**:
- The only option that satisfies the equation \( 56x + 72y = 8 \) is:
\[ x = 4, y = -3 \]

### Final Answer:
\[ x = 4, \quad y = -3 \]