Transport the property by positive multiplication

For every pair of natural numbers $n$ and $k$, if $k>0$ and there exist natural numbers $x,y,z$ such that $1\le x$, $x<y$, $y<z$, and $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, then there exist natural numbers $u,v,w$ such that $1\le u$, $u<v$, $v<w$, and $\frac{4}{kn}=\frac{1}{u}+\frac{1}{v}+\frac{1}{w}$. Every fraction and each equality here is interpreted in the rational numbers, with the natural numbers in it viewed as rational numbers. The quantification over $n$ includes $n=0$; in that case rational division gives $4/0=0$, whereas the three positive reciprocal terms in the hypothesis have a positive sum, so the hypothesis cannot hold. The assumption $k>0$ excludes $k=0$.