\[dc = {\color{red}d} \gcd({\color{green}a},{\color{blue}b}) = \left( \prod_{p \in \mathbb{P}} p^{\color{red}\delta_p} \right) \left( \prod_{p \in \mathbb{P}} p^{\min({\color{green}\alpha_p},{\color{blue}\beta_p})} \right) = \prod_{p \in \mathbb{P}} p^{\min({\color{red}\delta_p}+{\color{green}\alpha_p}, {\color{red}\delta_p}+{\color{blue}\beta_p})} = \gcd({\color{red}d}{\color{green}a},{\color{red}d}{\color{blue}b})\]