Counting Ordered Pairs of Positive Integers for the Equation (x/y 225/(xy) y/x)

Understanding and solving the given equation (frac{x}{y} frac{225}{xy} frac{y}{x}) is essential for comprehending the underlying structure of the equation and the properties of its solutions. This article delves into the step-by-step process and the mathematical reasoning required to find the positive integer solutions for this equation.

Solving the Equation

To solve the equation (frac{x}{y} frac{225}{xy} frac{y}{x}), we begin by eliminating the fractions through multiplication by (xy).

Starting with: [ frac{x}{y} frac{225}{xy} frac{y}{x} ] Multiply every term by (xy) to eliminate the fractions: [ x^2 225 y^2 ] Rearrange the equation: [ x^2 - y^2 225 ] Factorize the left-hand side: [ (x - y)(x y) 225 ] This can be rewritten as: [ x^3 - 225x - y^2 0 ] Treating this as a quadratic in (y): [ y^2 x^3 - 225x ] For (y) to be a positive integer, (x^3 - 225x) must be a perfect square. Let: [ k^2 x^3 - 225x ] This leads to: [ x^3 - 225x - k^2 0 ] Now, we analyze the function (f(x) x^3 - 225x) and find values of (x) that yield perfect squares.

Finding Integer Solutions

Evaluating (f(x)) for various values of (x): [ begin{aligned} x 1: f(1) 1 - 225 -224 quad text{(not a perfect square)} x 2: f(2) 8 - 450 -442 quad text{(not a perfect square)} x 3: f(3) 27 - 675 -648 quad text{(not a perfect square)} x 15: f(15) 3375 - 3375 0 quad k 0 quad text{(valid solution)} x 16: f(16) 4096 - 3600 496 quad text{(not a perfect square)} x 20: f(20) 8000 - 4500 3500 quad text{(not a perfect square)} x 25: f(25) 15625 - 5625 10000 quad k 100 quad text{(valid solution)} end{aligned} ] Checking further values, we find: [ begin{aligned} x 15: y text{ can only be } 0 , text{not valid} x 25: y 100 , text{valid} end{aligned} ] Other (x) values either do not yield perfect squares or result in negative values.

Summary of Possible Solutions

After checking integers up to around 30, the valid solutions are: [ begin{aligned} x 15: y text{ can only be } 0 , text{not valid} x 25: y 100 , text{valid with pairs } (25,100), , (100,25) end{aligned} ] These pairs are the only valid solutions for the equation. Therefore, there are exactly two ordered pairs of positive integers (x, y) that satisfy the given equation:

Conclusion

Thus, the total number of ordered pairs of positive integers (x, y) that satisfy the equation is: [ boxed{2} ] This detailed exploration and step-by-step solving process reveal the unique nature of the equation and the constraints on its solutions.