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Solve the following equation, where a, b, and c are natural numbers.

 

\(a + \frac{1}{b + \frac{1}{c}} = \frac{10}{7}\)

 Feb 27, 2021
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Since $a,b,c$ all have to be natural numbers, $a=1$ because it must be less than $\frac{10}{7}$, but also an integer greater than $0$. Furthermore, we now have $$\frac{1}{b+\frac{1}{c}} = \frac{3}{7}$$ $$\frac{c}{bc+1} = \frac{3}{7}$$. Therefore, we find that $c=3$ and $b=2$. Thus, our answer is $(a,b,c) = \boxed{(1,2,3)}.$

 Feb 27, 2021

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