Question

Suppose one wishes to find distinct positive integers \( x, y \) such that \( \frac{x + y}{xy} \) is also a positive integer. What can be said about the number of such valid pairs?

• A. This is never possible.
• B. This is possible and the pair (x,y) is unique.
• C. This is possible and there are finite such pairs.
• D. This is possible and there are infinite such pairs.

Hint:

When an expression reduces to a Diophantine form and no integers satisfy it under constraints (like distinctness), it's invalid.

Answer 1

The Correct Option is A

We are given that \( \frac{x + y}{xy} \) is a positive integer. Let this value be \( k \).
Then: \[ \frac{x + y}{xy} = k x + y = kxy \tag{1} \] Rewriting this: \[ kxy - x - y = 0 1 = kxy - x - y \tag{2} \] This is equivalent to solving: \[ kxy - x - y = 1 \] Try small positive integer values for \( x \), \( y \), and \( k \), but this equation never holds true for distinct positive integers.
For example, try \( x = 1, y = 1 \): \[ \frac{1 + 1}{1} = 2 \text{integer, but } x = y \text{ (not distinct)} \] Try \( x = 2, y = 1 \frac{3}{2} = 1.5 \) — not an integer.
This pattern continues. Every time we test for distinct \( x, y \), the result is either not an integer or violates the equation.
Conclusion: No such pair of distinct positive integers satisfies the condition. \[ \boxed{\text{Never possible}} \]