If p, q, r are positive integers such that \(p + \cfrac{1}{q + \cfrac{1}{r}} = \dfrac{25}{19}\), then find q.
+51 Since \(p\), \(q\), and \(r\) must all be positive, \(p = 1\) because if it was any higher, \(q\) and \(r\) would need to be negative to make equation true. So we have \(1+\frac{1}{q+\frac{1}{r}}=\frac{25}{19}\). So \(\frac{1}{q+\frac{1}{r}}=\frac{6}{19}= \frac{6}{6q+\frac{6}{r}}\), so \(19=6q+\frac{6}{r}\). The only possibility of \(q\) is \(\boxed{3}\), with \(r\) being \(6\).
