Coupled Sets

Let $S \subseteq \mathbb{N}^*$ have at least two elements. A set $T$ is called a coupled set of $S$ if:

1. $T \subseteq \mathbb{N}^*$ and $T$ has at least two elements; 2. for any $x, y \in S$ with $y \neq x$, we have $xy \in T$; 3. for any $x, y \in T$ with $y > x$, we have $\dfrac{y}{x} \in S$.

Questions:

1. If $S_1 = \{1, 2, 4\}$, find the coupled set $T_1$ of $S_1$. 2. Suppose $S_2 = \{p_1, p_2, p_3, p_4\}$ with $p_4 > p_3 > p_2 > p_1$ has a coupled set $T_2$. Prove that for any $1 \leqslant i < j \leqslant 4$, we have $\dfrac{p_j}{p_i} \in S_2$. 3. Let $S = \{p_1, p_2, p_3, p_4\}$ with $p_4 > p_3 > p_2 > p_1 \geqslant 2$. Find the number of elements of a coupled set $T$ of $S$.