Do You Need Extrema For This?

Let $13x=17y=19z=a$ . We find that $a$ must be the "Least Common Multiple (LCM)" of $13$ , $17$ , $19$ . We know that $LCM(13,17,19)$ is simply $(13\times 17\times 19)$ , since $(13,17,19)$ are relatively prime.Thus we have these equalities:

$\begin{aligned} 13x &=13\times 17\times 19 \Rightarrow x = 17\times 19 =\boxed{323} \\ 17y&=13\times 17\times 19 \Rightarrow y = 13\times 19 =\boxed{247} \\ 19z&=13\times 17\times 19 \Rightarrow z = 13\times 17 =\boxed{221} \\ min\{x+y+z\}&=323+247+221=\large{\boxed{791}} \end{aligned}$

Relevant wiki: Lowest Common Multiple

We are given the equation, $13x=17y=19z$ , where $x,y$ and $z$ are natural numbers. Clearly, the minimum value of the sides of the equation is the LCM of $13,17$ and $19$ , which is $4199$ .

Therefore, lowest possible values of individual variables:

$\implies x = \frac{4199}{13} = \boxed{323}$

$\implies y = \frac{4199}{17} = \boxed{247}$

$\implies z = \frac{4199}{19} = \boxed{221}$

Then, minimum value of $x+y+z= \boxed{791}$