If `xsqrt(147) = ysqrt(507)`, where x and y are co-prime numbers , then the value of `(x-y)` is

Options

A. 6
B. 7
C. 8
D. 9

Answer

To solve the equation \( x \sqrt{147} = y \sqrt{507} \) where \( x \) and \( y \) are co-prime numbers, we can follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ x \sqrt{147} = y \sqrt{507} \]

### Step 2: Rearrange the equation
We can rearrange this equation to express the ratio of \( x \) and \( y \):
\[ \frac{x}{y} = \frac{\sqrt{507}}{\sqrt{147}} \]

### Step 3: Simplify the square roots
Next, we simplify the square roots:
\[ \sqrt{507} = \sqrt{3 \times 169} = \sqrt{3} \times 13 \]
\[ \sqrt{147} = \sqrt{3 \times 49} = \sqrt{3} \times 7 \]

### Step 4: Substitute back into the ratio
Now substituting back, we have:
\[ \frac{x}{y} = \frac{\sqrt{3} \times 13}{\sqrt{3} \times 7} \]
The \( \sqrt{3} \) cancels out:
\[ \frac{x}{y} = \frac{13}{7} \]

### Step 5: Identify values of \( x \) and \( y \)
From the ratio \( \frac{x}{y} = \frac{13}{7} \), we can conclude:
- \( x = 13k \)
- \( y = 7k \)

Where \( k \) is a positive integer. To ensure \( x \) and \( y \) are co-prime, \( k \) must be 1 (since any other value would introduce a common factor).

Thus, we have:
- \( x = 13 \)
- \( y = 7 \)

### Step 6: Calculate \( x - y \)
Now we can find \( x - y \):
\[ x - y = 13 - 7 = 6 \]

### Final Answer
The value of \( x - y \) is \( 6 \).

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