Show that $x|y \Rightarrow x \leq y$

Question

I have to show that $x|y \Rightarrow x \leq y$ where $x,y \in \mathbb{N} \land x,y \neq 0$

Can someone give me a start hint how I can show this? I guess I can proof by induction. Not quite sure where to start

$x|y \Leftrightarrow xn =y$

$\Rightarrow n=\frac{y}{x}\geq1 \Rightarrow y\geq x$ given your definition for x|y — wfw, Nov 27 '13 at 11:06

drhab · Accepted Answer · 2013-11-27T13:04:40.747

4

$y-x=xn-x=x\left(n-1\right)\geq0$

(Here $x,y\in\mathbb{N}\wedge x,y\neq0\wedge n\in\mathbb{Z}$ so $y=xn$ can only be true if $n$ is a positive integer.)

edited Nov 27 '13 at 13:04

answered Nov 27 '13 at 11:09

drhab

This looks pretty complete. Probably I need to specify that $n \in \mathbb{N}$ and $n \ge 1$ – Chris Nov 27 '13 at 11:49
Yes. For that have a look at my edit. – drhab Nov 27 '13 at 13:06

Martin Argerami · Answer 2 · 2013-11-27T16:54:02.510

4

You want to show that $nx\geq x$. This is simply the inequality $n\geq1$ multiplied on both sides by $x$.

edited Nov 27 '13 at 16:54

answered Nov 27 '13 at 11:10

Martin Argerami

2 Answers2