# [SOLVED] Show that r|b

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• Jan 21st 2010, 03:47 PM
Deepu
[SOLVED] Show that r|b
if r belongs integer and r is a non zerosolution of x^2+ax+b=0 (where a,b belongs to integers), prove that r|b.
• Jan 21st 2010, 03:56 PM
Jhevon
Quote:

Originally Posted by Deepu
if r belongs integer and r is a non zerosolution of x^2+ax+b=0 (where a,b belongs to integers), prove that r|b.

We are told that $\displaystyle r^2 + ar + b = 0$

$\displaystyle \Rightarrow r^2 + ar = -b$

$\displaystyle \Rightarrow r(r + a) = -b$

Now what can you say?

Spoiler:
Since $\displaystyle r,a \in \mathbb{Z}$, we have that $\displaystyle (r + a) \in \mathbb{Z}$, so that $\displaystyle r$ divides the left side of the equation. But this means it must also divide the right side. Hence, $\displaystyle r|b$
• Jan 21st 2010, 04:17 PM
Deepu
Quote:

Originally Posted by Jhevon
We are told that $\displaystyle r^2 + ar + b = 0$

$\displaystyle \Rightarrow r^2 + ar = -b$

$\displaystyle \Rightarrow r(r + a) = -b$

Now what can you say?

Spoiler:
Since $\displaystyle r,a \in \mathbb{Z}$, we have that $\displaystyle (r + a) \in \mathbb{Z}$, so that $\displaystyle r$ divides the left side of the equation. But this means it must also divide the right side. Hence, $\displaystyle r|b$

Thank you..
• Jan 21st 2010, 04:20 PM
Jhevon
Quote:

Originally Posted by Deepu
Thank you..

the conclusion follows even more immediately if you recall that $\displaystyle m|n \Longleftrightarrow n = mk \text{ for some }k \in \mathbb{Z}$