<s>[INST] Solution Of Diophantine Equations 
Find all integer solutions for the equation:
  $$(x+y)(y+z)(z+x)=txyz$$
  such that gcd$(x, y)=1$, gcd$(y, z)=1$ and gcd$(z, x)=1$

Now we can write our equation as
$$(\frac{x+y}{y})(\frac{y+z}{z})(\frac{x+z}{x})=t$$
This gives $$(1+\frac{x}{y})(1+\frac{y}{z})(1+\frac{z}{x})=t$$
Now as $t$ is an integer and gcd$(x, y)=1$, gcd$(y, z)=1$ and gcd$(z, x)=1$ so the only possibility is $$x=y=z=1$$ giving a solution as $$(x, y, z, t)=(1, 1, 1, 8)$$
Am I Right?
 [/INST] Hint:
Say $x,y,z$ are positive. We have $x|y+z$ and $y|x+z$ and $z|x+y$. We can assume $x\geq y\geq z$. Then $y+z= kx$ where $k=1$ or $k=2$.
If k=1we have $(2y+z)(2z+y)=tyz$ and so $y|2z^2$ so $y=1$ or $y=2$.
If k=2 we have $(3y+z)(3z+y) =2tyz$ so $y| 3z^2$ so $y=1$ or $y=3$...
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