<s>[INST] let $x ,y ,z \in \mathbb{N} ,x \leq y\leq z$ and : $\frac{100}{336}=\frac{1}{x}+\frac{1}{xy}+\frac{1}{xyz}$then : $x+y+z =?$ let $x ,y ,z \in \mathbb{N} ,x \leq y\leq z$
and :
$$\frac{100}{336}=\frac{1}{x}+\frac{1}{xy}+\frac{1}{xyz}$$
then :
$$x+y+z =?$$
my try :
$$\frac{100}{336}=\frac{1}{x}+\frac{1}{xy}+\frac{1}{xyz}=\frac{xy+z+1}{xyz}=\frac{25}{84}$$
now ?
 [/INST] It should be $x=4$, $y=6$, $z=7$ and the answer to the original question is 17.
You can use the method described here for obtaining the Engel expansion of a number.
The Engel expansion for $x$ is written $x=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\cdots$
As described in the wiki, the non-decreasing values $a_1, a_2, \ldots$ can be obtained using 
$$u_1=x
\qquad a_k=\left \lceil \frac{1}{u_k} \right \rceil
\qquad u_{k+1}=u_ka_k-1
$$
and stopping when we reach $u_k = 0$.
Starting with $x=\frac{25}{84}$ leads to the required solution.
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