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 \( y \) in the equation \( \text{H.C.F.}(56, 72) = 56x + 72y \).

### Step 1: Find the H.C.F. of 56 and 72

To find the H.C.F. (Highest Common Factor) of 56 and 72, we can use the Euclidean algorithm.

1. Divide 72 by 56:
\[ 72 = 56 \times 1 + 16 \]

2. Now, take the remainder (16) and divide 56 by 16:
\[ 56 = 16 \times 3 + 8 \]

3. Next, take the remainder (8) and divide 16 by 8:
\[ 16 = 8 \times 2 + 0 \]

Since the remainder is now 0, the last non-zero remainder is the H.C.F.:
\[ \text{H.C.F.}(56, 72) = 8 \]

### Step 2: Express the H.C.F. in the form \( 56x + 72y \)

Now we need to express 8 as a linear combination of 56 and 72.

From the previous steps, we have:
1. From \( 56 = 16 \times 3 + 8 \), we can express 8 as:
\[ 8 = 56 - 16 \times 3 \]

2. We also know from the first division that \( 16 = 72 - 56 \times 1 \). Substitute this into the equation for 8:
\[ 8 = 56 - 3(72 - 56 \times 1) \]
Simplifying this gives:
\[ 8 = 56 - 3 \times 72 + 3 \times 56 \]
\[ 8 = 4 \times 56 - 3 \times 72 \]

### Step 3: Identify the values of \( x \) and \( y \)

Now we can compare this with the equation \( 8 = 56x + 72y \):
\[ 8 = 56(4) + 72(-3) \]

From this, we can see that:
- \( x = 4 \)
- \( y = -3 \)

### Final Answer
Thus, the values of \( x \) and \( y \) are:
\[ x = 4, \quad y = -3 \]

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