Question

The total number of integer pairs \( (x, y) \) satisfying the equation \( x + y = xy \) is

• A. 0
• B. 1
• C. 2
• D. None of the above

Hint:

When faced with Diophantine equations, try rewriting the equation in factored form to identify possible integer solutions.

Answer 1

The Correct Option is B

Rewriting the equation \( x + y = xy \), we get: \[ xy - x - y = 0 \] Adding 1 to both sides: \[ xy - x - y + 1 = 1 \] Factoring: \[ (x - 1)(y - 1) = 1 \] Thus, the integer pairs \( (x - 1) \) and \( (y - 1) \) are factors of 1. The only pairs of integers that satisfy this are \( (x - 1, y - 1) = (1, 1) \), which gives \( (x, y) = (2, 2) \). \[ \boxed{1} \]