A logarithmic equation

Algebra Level 5

The sum of all (positive) solutions of the equation

\[\log_{16}x +\log _x 16=\log_{512} x + \log_x {512}\]

can be written as \( \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. What are the last three digits of \(a+b\)?

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