Croatian Junior Mathematical Olympiad

Let $d$ be the greatest common divisor of $m$ and $n$, i.e. $m=d{m}^{\prime }$ and $n=d{n}^{\prime }$, where $m^{\prime }$ and $n^{\prime }$ are relatively prime positive integers of different parity. Now we need to prove that $$\frac{3d^2{m}^{\prime 2}+5d^2{m}^{\prime }{n}^{\prime }}{3d^2{n}^{\prime 2}+d{m}^{\prime }{n}^{\prime }}=\frac{m^{\prime }(3m^{\prime }+5n^{\prime })}{n^{\prime }(m^{\prime }+3n^{\prime })}$$ is not a positive integer. Note that $3m^{\prime }+5n^{\prime }$ and $m^{\prime }+3n^{\prime }$ are both odd. If $m^{\prime }$ is odd and $n^{\prime }$ is even, the even $n^{\prime }(m^{\prime }+3n^{\prime })$ clearly cannot divide the odd $m^{\prime }(3m^{\prime }+5n^{\prime })$. Otherwise, let the odd $k$ be the greatest common divisor of $3m^{\prime }+5n^{\prime }$ and $m^{\prime }+3n^{\prime }$. Then we have $$\begin{array}{rlrl}k& \mid 3\cdot (3m^{\prime }+5n^{\prime })-5\cdot (m^{\prime }+3n^{\prime }),& \text{ i.e. }k& \mid 4m^{\prime },\\ k& \mid 1\cdot (3m^{\prime }+5n^{\prime })-3\cdot (m^{\prime }+3n^{\prime }),& \text{ i.e. }k& \mid -4n^{\prime },\end{array}$$ from which it follows that $k$ divides both $m^{\prime }$ and $n^{\prime }$, so $k=1$ and both factors in $$\frac{m^{\prime }}{n^{\prime }}\cdot \frac{3m^{\prime }+5n^{\prime }}{m^{\prime }+3n^{\prime }}$$ are irreducible fractions. Since $m^{\prime }+3n^{\prime }>m^{\prime }$, the proof is finished.