Find all positive integers of x and y where equation is 1/sqrt x+1/sqrt y =1/sqrt 20?

Answer

To solve the equation \( \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} = \frac{1}{\sqrt{20}} \) for positive integers \( x \) and \( y \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} = \frac{1}{\sqrt{20}} \]
We can express \( \sqrt{20} \) as \( 2\sqrt{5} \):
\[ \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} = \frac{1}{2\sqrt{5}} \]

### Step 2: Clear the fractions
To eliminate the fractions, multiply through by \( 2\sqrt{5}\sqrt{x}\sqrt{y} \):
\[ 2\sqrt{5}\sqrt{y} + 2\sqrt{5}\sqrt{x} = \sqrt{x}\sqrt{y} \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ \sqrt{x}\sqrt{y} - 2\sqrt{5}\sqrt{x} - 2\sqrt{5}\sqrt{y} = 0 \]

### Step 4: Substitute variables
Let \( \sqrt{x} = m \) and \( \sqrt{y} = n \). Then, we can rewrite the equation as:
\[ mn - 2\sqrt{5}m - 2\sqrt{5}n = 0 \]

### Step 5: Factor the equation
Rearranging gives:
\[ mn - 2\sqrt{5}m - 2\sqrt{5}n + 4 = 4 \]
This can be factored as:
\[ (m - 2\sqrt{5})(n - 2\sqrt{5}) = 4 \]

### Step 6: Find integer solutions
Now we need to find pairs of integers \( (m - 2\sqrt{5}) \) and \( (n - 2\sqrt{5}) \) that multiply to 4. The factor pairs of 4 are:
- \( (1, 4) \)
- \( (2, 2) \)
- \( (-1, -4) \)
- \( (-2, -2) \)

### Step 7: Solve for \( m \) and \( n \)
For each factor pair, we solve for \( m \) and \( n \):
1. For \( (1, 4) \):
- \( m - 2\sqrt{5} = 1 \) → \( m = 1 + 2\sqrt{5} \) (not an integer)
- \( n - 2\sqrt{5} = 4 \) → \( n = 4 + 2\sqrt{5} \) (not an integer)

2. For \( (2, 2) \):
- \( m - 2\sqrt{5} = 2 \) → \( m = 2 + 2\sqrt{5} \) (not an integer)
- \( n - 2\sqrt{5} = 2 \) → \( n = 2 + 2\sqrt{5} \) (not an integer)

3. For \( (-1, -4) \):
- \( m - 2\sqrt{5} = -1 \) → \( m = -1 + 2\sqrt{5} \) (not an integer)
- \( n - 2\sqrt{5} = -4 \) → \( n = -4 + 2\sqrt{5} \) (not an integer)

4. For \( (-2, -2) \):
- \( m - 2\sqrt{5} = -2 \) → \( m = -2 + 2\sqrt{5} \) (not an integer)
- \( n - 2\sqrt{5} = -2 \) → \( n = -2 + 2\sqrt{5} \) (not an integer)

### Step 8: Check for integer values
We can also try integer values for \( m \) and \( n \) directly. We find:
- If \( m = 4 \) and \( n = 4 \), then \( x = 5 \cdot 4^2 = 80 \) and \( y = 5 \cdot 4^2 = 80 \).
- If \( m = 3 \) and \( n = 6 \), then \( x = 5 \cdot 3^2 = 45 \) and \( y = 5 \cdot 6^2 = 180 \).
- If \( m = 6 \) and \( n = 3 \), then \( x = 5 \cdot 6^2 = 180 \) and \( y = 5 \cdot 3^2 = 45 \).

### Final Solutions
The positive integer pairs \( (x, y) \) that satisfy the equation are:
- \( (80, 80) \)
- \( (45, 180) \)
- \( (180, 45) \)