Question 207, Find all positive integers that can be written in the form $\frac{[x, y]+[y, z]}{[x, z]}$, where $x, y, z$ are three positive integers, and $[a, b]$ denotes the least common multiple of the positive integers $a, b$.
Question 207: Proof: Let $(x, y, z) = d_{0}, \left(\frac{y}{d_{0}}, \frac{z}{d_{0}}\right) = d_{1}, \left(\frac{x}{d_{0}}, \frac{z}{d_{0}}\right) = d_{2}, \left(\frac{x}{d_{0}}, \frac{y}{d_{0}}\right) = d_{3}$. Let $x_{1} = \frac{x}{d_{0} d_{2} d_{3}}, y_{1} = \frac{y}{d_{0} d_{1} d_{3}}, z_{1} = \frac{z}{d_{0} d_{1} d_{2}}$, then $x_{1}, y_{1}, z_{1}$ are pairwise coprime positive integers, and
\[
[x, y] = d_{0} d_{1} d_{2} d_{3} \cdot x_{1} y_{1}, [y, z] = d_{0} d_{1} d_{2} d_{3} \cdot y_{1} z_{1}, [x, z] = d_{0} d_{1} d_{2} d_{3} \cdot x_{1} \hat{z}_{1}
\]
Thus,
\[
\frac{[x, y] + [y, z]}{[x, z]} = \frac{x_{1} y_{1} + y_{1} z_{1}}{x_{1} z_{1}} = \frac{y_{1} (x_{1} + z_{1})}{x_{1} z_{1}} \in \mathbb{Z}
\]

Noting that $x_{1}, y_{1}, z_{1}$ are pairwise coprime positive integers, it follows that $x_{1} = z_{1} = 1$, hence $\frac{[x, y] + [y, z]}{[x, z]} = 2 y_{1}$ is a positive even number.
Conversely, for any positive even number $2m$, take $x = z = 1, y = m$, then $\frac{[x, y] + [y, z]}{[x, z]} = 2m$. In summary, the positive integers that can be expressed in the form $\frac{[x, y] + [y, z]}{[x, z]}$ are all positive even numbers.
Note: The following Venn diagram provides a more intuitive representation:

(Note: The Venn diagram is not provided in the text, but it is mentioned for visual aid.)