# Prove that the equation xy = 2006 (x + y) has integer solutions.

July 31, 2021 | education

| Let’s transform this equation:

xy = 2006 (x + y);

xy – 2006x – 2006y = 0;

Add 2006 ^ 2 to the left and right sides;

xy – 2006x – 2006y + 2006 ^ 2 = 2006 ^ 2;

Factor the left side to get the equation:

(y – 2006) (x – 2006) = 2006 ^ 2;

This equation has an integer solution, for example:

x = 4012; y = 4012 – is the solution to this equation.

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