Question 176
Let $S$ be a set of positive integers such that
$$
\lfloor\sqrt{x}\rfloor=\lfloor\sqrt{y}\rfloor \text{ for all } x, y \in S
$$
Prove that if $x, y, z, t \in S$ with $(x, y) \neq(z, t)$ and $(x, y) \neq(t, z)$, then $x y \neq z t$.
($\lfloor.\rfloor$ denotes the integer part.)
$$
\lfloor\sqrt{x}\rfloor=\lfloor\sqrt{y}\rfloor \text{ for all } x, y \in S
$$
Prove that if $x, y, z, t \in S$ with $(x, y) \neq(z, t)$ and $(x, y) \neq(t, z)$, then $x y \neq z t$.
($\lfloor.\rfloor$ denotes the integer part.)