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# help

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If p, q, r are positive integers such that $$p + \cfrac{1}{q + \cfrac{1}{r}} = \dfrac{25}{19}$$, then find q.

Jun 12, 2020

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 $$6$$.