Question 168
a. Find two quadruples of positive integers $(a, b, c, n)$, each with a different value of $n$ greater than $3$, such that
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = n
$$
b. Show that if $a, b, c$ are nonzero integers such that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer, then $a b c$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = n
$$
b. Show that if $a, b, c$ are nonzero integers such that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer, then $a b c$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)