Similarly, by exploiting the set $\mathcal{C}_n^{\prime} = \left\{ \left( d_1, d_2, d_3 \right) \in \left( \mathbb{N}^* \right)^3 \mid d_1 d_2 d_3 = n \right\}$, show that $*$ is associative.
Similarly, by exploiting the set $\mathcal{C}_n^{\prime} = \left\{ \left( d_1, d_2, d_3 \right) \in \left( \mathbb{N}^* \right)^3 \mid d_1 d_2 d_3 = n \right\}$, show that $*$ is associative.