<s>[INST] Solving $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$, $\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$ Whether non-zero integers $a, b, c$ with the property that $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$$ and $$\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$$ Calculate all possible values for $m + n$. 
 [/INST] Let 
$\frac{a}{b}=x$ , $\frac{b}{c}=y$ , $\frac{c}{a}=z$
$x+y+z=m$, $xy+yz+zx=n$
$m+n+xyz+1=1+x+y+z+xy+yz+zx+xyz$
$m+n+2=(x+1)(y+1)(z+1)\geqslant 2^3\sqrt{xyz}=2^3$ AM-GM
$m+n+2\geqslant 8$
$m+n \geqslant 6 $
I'm not sure about my solution
</s>