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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