Suppose that $x$, $y$ and $z$ are relatively prime integers and $m$ be a positive integer. Solve the equation $$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = m$$ It is true to set x=y (WLOG) and then attemp to solve it?
Asked
Active
Viewed 102 times
1
-
in pairs not as a triple. If it has not any solution how can we show that? – Apr 21 '18 at 10:18
-
Useful link: https://math.stackexchange.com/questions/848718/integer-values-of-fracxy-fracyz-fraczx for very similar problem. – Oleg567 Apr 21 '18 at 11:03
-
2Possible duplicate of Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$? – user Apr 21 '18 at 13:16
1 Answers
1
I am assuming the pairwise relatively prime integers $x, y, z$. Multiply both sides by $xyz$ to get $x^2z+y^2x+z^2y=mxyz$. The RHS is divisible by $x$. Hence, $x$ divides the LHS as well. Thus, $x$ divides $z^2y$. Since, $x$ is relatively prime with $z^2y$, we get $x=1$. Analogously, $y=z=1$. Thus, $m=3$.
