For all $A, B \in\Bbb Z$, if $A\mid B$ and $B\mid A$, then $A =B$.

Question

I know this is a simple question, but I am struggling a bit. What I have so far is. Let $A, B$ be arbitrary elements in $\Bbb Z$ and assume $A\mid B$ and $B\mid A$ is true. $A=B \Rightarrow \dfrac{A}{B}=\dfrac{B}{B} \Rightarrow \dfrac{A}{B} =1$ and $A=B \Rightarrow \dfrac{A}{A}=\dfrac{B}{A}\Rightarrow 1=\dfrac{B}{A}$. Therefore $A$ and $B$ are elements in the set and are equal to each other.

Does this prove the statement?

or more generally $A=-B\not=0$ – Henry Oct 09 '20 at 01:55 — Henry, Oct 09 '20 at 01:55
@KentaS I want to prove that A=1 and B=1 – Oct 09 '20 at 01:57 — , Oct 09 '20 at 01:57
@bhhha1232 I am saying your statement is false. – Kenta S Oct 09 '20 at 01:58 — Kenta S, Oct 09 '20 at 01:58
@KentaS and that is because it is the set of integers? – Oct 09 '20 at 02:01 — , Oct 09 '20 at 02:01
@bhhha1232 That is because I found a counterexample. – Kenta S Oct 09 '20 at 02:01 — Kenta S, Oct 09 '20 at 02:01

score 1 · Answer 1 · answered Oct 09 '20 at 01:58

1

We have $a,b\in\Bbb Z\setminus\{0\}$. If $a\mid b$ then $a=rb$ for some $r\in\Bbb Z$. If $b\mid a$ then $b=sa$ for some $s\in\Bbb Z$. Then $a=rb=rsa\Rightarrow rs=1\Rightarrow r,s=\pm1$, so $a=\pm b$.

answered Oct 09 '20 at 01:58

Darsen

3,549

Does this answer your question? – Darsen Oct 28 '20 at 17:22

For all $A, B \in\Bbb Z$, if $A\mid B$ and $B\mid A$, then $A =B$.

1 Answers1