# Simple problem. Did I get it right?

• Aug 26th 2008, 10:55 AM
espen180
Simple problem. Did I get it right?
If $\displaystyle a|b$ and $\displaystyle b|a$, is $\displaystyle a=b$?

I did as follows: If $\displaystyle a|b$, there is a number $\displaystyle r$ in $\displaystyle \mathbb{Z}$ such that $\displaystyle b=r\cdot a$.
Likewise, if $\displaystyle b|a$, there is a number $\displaystyle s$ in $\displaystyle \mathbb{Z}$ such that $\displaystyle a=s\cdot b$.

This shows us that $\displaystyle a=r\cdot s \cdot b$ and $\displaystyle b=r\cdot s \cdot a$. Because of this, $\displaystyle r=s^{-1}$.
$\displaystyle r$ and $\displaystyle s$ er both in $\displaystyle \mathbb{Z}$, therefore $\displaystyle r=s=1$ and $\displaystyle \underline{\underline{a=b}}$

Was my reasoning here correct, and did I get the right answer?
• Aug 26th 2008, 11:01 AM
ThePerfectHacker
No, $\displaystyle a = |b|$.
• Aug 26th 2008, 12:21 PM
Opalg
Quote:

Originally Posted by ThePerfectHacker
No, $\displaystyle a = |b|$.

No, |a| = |b| (Bigsmile)