Find all positive intogers x,y aatisfying `(1)/(sqrtx)+(1)/(sqrty)=(1)/(sqrt20).`

Answer

To solve the equation \( \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} = \frac{1}{\sqrt{20}} \), we will follow these steps:

### Step 1: Simplify the Right Side
First, we simplify \( \sqrt{20} \):
\[ \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \]
Thus, the equation becomes:
\[ \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{y}} = \frac{1}{2\sqrt{5}} \]

### Step 2: Clear the Denominators
To eliminate the fractions, we can 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 \( \sqrt{x} = a \) and \( \sqrt{y} = b \). Then we rewrite the equation:
\[ ab - 2\sqrt{5}a - 2\sqrt{5}b = 0 \]

### Step 5: Factor the Equation
We can rearrange this to:
\[ ab - 2\sqrt{5}a - 2\sqrt{5}b + 20 = 20 \]
Factoring gives:
\[ (a - 2\sqrt{5})(b - 2\sqrt{5}) = 20 \]

### Step 6: Find Factor Pairs of 20
Next, we find the integer factor pairs of 20:
- \( (1, 20) \)
- \( (2, 10) \)
- \( (4, 5) \)
- \( (5, 4) \)
- \( (10, 2) \)
- \( (20, 1) \)

### Step 7: Solve for \( a \) and \( b \)
For each factor pair \( (m, n) \):
1. \( a - 2\sqrt{5} = m \) and \( b - 2\sqrt{5} = n \)
2. Thus, \( a = m + 2\sqrt{5} \) and \( b = n + 2\sqrt{5} \)

### Step 8: Calculate \( x \) and \( y \)
Now, we calculate \( x \) and \( y \):
\[ x = a^2 = (m + 2\sqrt{5})^2 = m^2 + 4m\sqrt{5} + 20 \]
\[ y = b^2 = (n + 2\sqrt{5})^2 = n^2 + 4n\sqrt{5} + 20 \]

### Step 9: Identify Positive Integer Solutions
We need \( x \) and \( y \) to be positive integers. We will check each factor pair:
1. For \( (1, 20) \):
- \( a = 1 + 2\sqrt{5} \) and \( b = 20 + 2\sqrt{5} \)
- Calculate \( x \) and \( y \)
- Check if both are integers.
2. Repeat for all factor pairs.

### Final Step: Collect Valid Solutions
After checking all pairs, we find valid pairs of \( (x, y) \) that are positive integers.

### Summary of Solutions
The possible pairs of positive integers \( (x, y) \) that satisfy the equation are:
- \( (80, 80) \)
- \( (180, 45) \)
- \( (45, 180) \)