If `x sqrt(243) = y sqrt(867)`, where x and y are coprime numbers, then the value of (x-y) is

Options

A. 3
B. 8
C. 4
D. 6

Answer

To solve the equation \( x \sqrt{243} = y \sqrt{867} \) where \( x \) and \( y \) are coprime numbers, we can follow these steps:

### Step 1: Simplify the square roots
First, we need to simplify \( \sqrt{243} \) and \( \sqrt{867} \).

1. **Finding \( \sqrt{243} \)**:
- \( 243 = 3^5 \)
- Therefore, \( \sqrt{243} = \sqrt{3^5} = 3^{5/2} = 9\sqrt{3} \).

2. **Finding \( \sqrt{867} \)**:
- We can factor \( 867 \) into its prime factors:
- \( 867 = 3 \times 289 \)
- \( 289 = 17^2 \)
- Thus, \( \sqrt{867} = \sqrt{3 \times 17^2} = 17\sqrt{3} \).

### Step 2: Rewrite the equation
Now we can rewrite the original equation using the simplified square roots:
\[ x \cdot 9\sqrt{3} = y \cdot 17\sqrt{3} \]

### Step 3: Cancel out the common term
Since \( \sqrt{3} \) is common on both sides, we can cancel it out:
\[ x \cdot 9 = y \cdot 17 \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ \frac{x}{y} = \frac{17}{9} \]

### Step 5: Express \( x \) and \( y \) in terms of a common variable
Let \( x = 17k \) and \( y = 9k \) for some integer \( k \).

### Step 6: Ensure \( x \) and \( y \) are coprime
Since \( x \) and \( y \) must be coprime, the only value for \( k \) that satisfies this is \( k = 1 \):
- Thus, \( x = 17 \) and \( y = 9 \).

### Step 7: Calculate \( x - y \)
Now we can find \( x - y \):
\[ x - y = 17 - 9 = 8 \]

### Final Answer
The value of \( x - y \) is \( 8 \).

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