divisibility question: if two integers can both divide each other, do they have to be equal?

Question

if x ,y ∈ Z. and x|y,y|x,then x does NOT equal to y. Can anyone give me a counter example please?

Answer 1

For general integers $x, y$, if $x | y$ and $y | x$ then you can't quite conclude $x = y$. But the following holds: $$ \forall x, y \in \mathbb{Z} \;:\; (x \mid y \text{ and } y \mid x) \implies |x| = |y|. \tag{1} $$

In particular, if $x \ge 0$ and $y \ge 0$, then $x | y$ and $y | x$ implies $x = y$.

Answer 2

For positive integers $x$ and $y$ if $x|y$ and $y|x$ then $x=y$.

Proof:

By prime factorization, let $x = \prod_p p^{x_p}$ and $y = \prod_p p^{y_p}$ .

If $x|y$ then $x_p \le y_p$. If $y|x$ then $y_p \le x_p$.

Therefore, if $x|y$ and $y|x$ then $x_p \le y_p$ and $y_p \le x_p$, so $x_p = y_p$ so that $x = y$.

Answer 3

A really trivial proof for positive $x$ and $y$.

If $x|y$, then $y = ax$ where $a \ge 1$.

If $y|x$, then $x = by$ where $b \ge 1$.

Therefore $x = by =bax $ so $ba = 1$. Since $a\ge 1$ and $b \ge 1$, we must have $a = b = 1$ so $x = y $.