IMO 2007 Shortlist

For natural numbers \(x\) and \(y\), let \(\gcd(x, y)\) denote the greatest common divisor of \(x\) and \(y\). How many pairs of natural numbers \(x\) and \(y\) such that they satisfy the equation below?

\[xy = x + y + \gcd (x, y)\]

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