MathNet 2019-part-01 — 184 problems (EN–VI) Question 29

Determine the smallest possible positive integer $n$ with the following property: For all positive integers $x$, $y$ and $z$ with $x \mid y^3$ and $y \mid z^3$ and $z \mid x^3$ we also have $xyz \mid (x+y+z)^n$.
(Gerhard J. Woeginger)