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Stuck on proof – Theorem 7, Ch 5.8.1 in Ross’s Intro to Algebraic Number Theory
Here is the problem:
I want to prove that the $x_i$ are pairwise relatively prime. I have the basic outline of the proof in my head, but the idea is very foggy. I can’t wrap my head around the idea of showing that it is true for all $x_i$ in the set $S$, then taking the union of those sets. I know that there is a theorem that says if $x_1x_2\cdots x_k = y_1y_2\cdots y_l$ with positive integers $x_i$ and $y_j$ and $k\lt l$, then $x_i$ and $y_j$ are relatively prime, but I need to prove the result for a single $x_i$ and a single $y_j$.
Could someone explain the basic idea? If I knew what $S$ was, I could build up an intuitive feel for the proof, but the $x_i$ are not given to me. I know it’s something like a basic gcd computation between them, but I can’t seem to make this geometric intuition work.
A:
$S=\{a_{1},a_{2},\ldots,a_{n}\}$ be $n$ distinct positive integers.
$S$ has $n$ elements. $A=\{a_{1},a_{2},\ldots,a_{n}\}$ be a subset of $S$.
First observe