Firefighting | Math Problem

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Seeking attentions of @gonitzoggoteam
(9,13) is not the only solution. For the sake of minimization, we can assume that \frac{m}{k} \approx\frac{49}{34}. Continued fraction decomposition of \frac{49}{34} gives:

\frac{49}{34}=1+\frac{15}{34} =1+\frac{1}{\frac{34}{15}} =1+\frac{1}{2+\frac{4}{15}} =\cdots \cdots \cdots =1+\frac{1}{2+\frac{1}{3+\frac{1}{1+\frac{1}{3}}}}
Now , for better approximation, replace the \frac{1}{3} at the very end part of the continued fraction with \frac{1}{2} to get \frac{m}{k}=\frac{36}{25} giving (25,36) as a valid solution for (k.m). To verify the answer, check if |(49k-34m)| is minimized with (25,36)