<s>[INST] Find all positive integer roots of $\,5xy=19x+96y$ 
Find all positive integer roots of : $5xy=19x+96y$

I tried using decomposition technique but no success...,it seems suitable  factorization of this equation is IMPOSSIBLE!!
Handy calculations show that it has as many as 6 answers.
 [/INST] We have $19x=(5x-96)y$, so $y=0\bmod19$ or $x=4\bmod19$.
If $y=19n$, then we have $\frac{96}{x}=5-\frac{1}{n}$. So $x\le24$, and $x(5n-1)=96n\ (*)$. Checking $n=1,2,3,4,5$ we find that $n=1$ works giving solution $(24,19)$. $n=2,3,4$ do not work. $n=5$ gives the solution $(20,95)$. If $n>5$ then since $5n-1,n$ are coprime we must have $5n-1|96$ and hence $5n-1=48,96$, neither of which work.
If $x=4\bmod19$, then $(5n-4)y=19n+4$. If $n=1$ we have $y=23$, giving the solution $(23,23)$. If $n\ge2$, then we have $5=\frac{96}{x}+\frac{19}{y}=\frac{1}{y}\left(\frac{96}{5n-4}+19\right)\le\frac{35}{y}$, so $y\le7$. If $y=7$, we get the solution $(42,7)$. If $y=6$ we get $11n=28$ which fails. If $y=5$ we get the solution $(80,5)$. If $y=4$ we get the solution $(384,4)$. If $y\le3$ we get $(19-5y)n=-4-4y$ which fails.
So in total we have the six solutions $(20,95),(23,23),(24,19),(42,7),(80,5),(384,4)$.
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