6. The greatest common divisor (GCD) between each pair of three numbers is $3$, $4$, and $5$. What is the minimum sum of these three numbers?

The greatest common divisor (GCD) between each pair of three numbers is $3$, $4$, and $5$. What is the minimum sum of these three numbers?
【Analysis】Let these three numbers be $\mathrm{A}, \mathrm{B}, \mathrm{C}$. Without loss of generality, we have:
$$
\left\{\begin{array} { l } 
{ ( \mathrm { A } , \mathrm { B } ) = 3 } \\
{ ( \mathrm { A } , \mathrm { C } ) = 4 } \\
{ ( \mathrm { B } , \mathrm { C } ) = 5 }
\end{array} \Rightarrow \left\{\begin{array}{l}
\mathrm{A}=12 \mathrm{a} \\
\mathrm{B}=15 \mathrm{~b}(\mathrm{a}, \mathrm{b}, \mathrm{c})=1 \\
\mathrm{C}=20 \mathrm{c}
\end{array}\right.\right.
$$

Therefore, the smallest values of these three numbers are: $12, 15, 20$, and their sum is 47.