1972 USAMO 第 1 题

The symbols $(a,b,\ldots,g)$ and $[a,b,\ldots, g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $a,b,\ldots, g$. For example, $(3,6,18)=3$ and $[6,15]=30$. Prove that

$$\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}$$

.

符号$(a,b,\ldots,g)$和$[a,b,\ldots, g]$分别表示正整数$a,b,\ldots, g$的最大公约数和最小公倍数。例如，$(3,6,18)=3$ 和 $[6,15]=30$。证明

$$\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}$$

。