5. Given positive integers $a, b, c$, define the function $f(x, y, z)=a x+b y+c z$, where $x, y, z \in \mathbf{Z}$. Try to find the smallest positive integer value of $f(x, y, z)$.
5. Let $f_{\min }=a x_{0}+b y_{0}+c z_{0}=d_{0}$. Suppose $(a, b, c)=d$, clearly $d \mid d_{0}$, thus $d \leqslant d_{0}$. On the other hand, by Bézout's theorem, there exist integers $x, y, z$, such that $a x+b y+c z=d$. Since $d_{0}$ is the smallest, we have $d_{0} \leqslant d$. Therefore, $d_{0}=d$, i.e., $f_{\min }=(a, b, c)$.