I am a high school student and came across this problem on Number Theory.
(1) Prove that for any given positive integer $n$, there exists a set $S \subseteq \mathbb{N^{*}}$, such that for all $1\le k \le n$, any subset $T$ of $S$ of size $k$, satisfies $\displaystyle \prod_{x\in T}x=t_x^k,t_x \in \mathbb{N}$.
(2) Prove that it is impossible to find an infitite set $S \subseteq \mathbb{N^{*}}$, such that for all positive integer $k$, any subset $T$ of $S$ of size $k$ satisfies $\displaystyle \prod_{x\in T}x=t_x^k,t_x \in \mathbb{N}$.
Note that $t_x$ can be different for diffrent $x$. Or as in Chinese, 任意 $k$ 元子集之积都是 $k$ 次方幂。
I can easily give a construction of the first question, which is to arbitrarily choose $n$ integers $a_1, a_2, \ldots, a_n$, and let $S=\{a_1^{n!}, a_2^{n!}, \ldots, a_n^{n!}\}$. However, I just cannot understand why I can find such a set of arbitrary size but cannot find such a set of infinite size.
Can anyone explain why to me in a high-schooler-understandable way?
update1: Chinese statement.
