From strict to non-strict

Algebra Level 4

Let \(a\) and \(b\) be positive integers such that \(\frac{ab}{a + b} > 2017.\) Let \(C\) be the greatest possible real number such that \(\frac{ab}{a + b} \geq C.\) If \(C\) can be written in the form \(\frac{p}{q},\) where \(p\) and \(q\) are coprime positive integers, find the last three digits of \(q.\)

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