Equal Logs

Algebra Level 4

\[\log_9 p = \log_{12} q = \log_{16} (p + q)\]

Suppose that \(p\) and \(q\) are positive numbers for which satisfy the equation above.

Then \(q/p\) can be expressed in the form \( \frac { x + \sqrt{y}}{z} \), where \(x, y, z\) are positive integers, and \(y\) is not divisible by the square of a prime. Find \(x + y + z\).

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