1. Given positive integers $a, b, c$ satisfy
$$
10 a^{2}-3 a b+7 c^{2}=0 \text {. }
$$

Find the minimum value of $(a, b)(b, c)(c, a)$.
1.3.
$$
\begin{array}{l} 
\text { Given } 10 a^{2}+7 c^{2} \equiv a^{2}+c^{2} \equiv 0(\bmod 3) \\
\Rightarrow 3|a, 3| c \Rightarrow(a, c) \geqslant 3 \\
\Rightarrow(a, b)(b, c)(c, a) \geqslant 3 .
\end{array}
$$

Also, when $a=c=3, b=17$, the minimum value 3 is achieved.