Prove $\exists r,s\in\Bbb{Z}$ such that $ar+ns=1\iff$ there exists an integer $b\ne 0$ such that $ab\equiv 1 \pmod n$

Asked Sep 02 '20 at 02:39

Active Sep 02 '20 at 02:54

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For this problem $a,b,n\in\Bbb{Z},n\ge 1$

Prove that there exist integers $r,s$ such that $ar+ns=1$ if and only if there exists a non-zero integer $b$ such that $ab\equiv 1 \pmod n$

So far this is my proof:

$(\impliedby)$ Assume $\exists$ a non-zero integer b such that $ab\equiv 1\pmod n$
(Want to show there exists integers $r,s$ such that $ar+ns = 1$).
By assumption, $n|ab-1\implies ab-1=ns$, where $s$ is an integer.

I am not sure what to do next I see that I can get the $1$ on the right hand side but how do I bring in $r$ in the equation. Also because it is a biconditional proof I know I need to show the other way as well but I am stuck.

edited Sep 02 '20 at 02:54

Fawkes4494d3

2,984

asked Sep 02 '20 at 02:39

NaturalMathLover

How to bring $r$ in: take $r$ as $b$ – J. W. Tanner Sep 02 '20 at 02:42
Please use MathJax, not whatever it is you are using. Here is a tutorial. Use \pmod{n} to get the parenthetical “mod n” with proper spacing. – Arturo Magidin Sep 02 '20 at 02:43
Thank you, but now I am left with ar - 1 =ns which then can be written as ar-ns=1. I need for it to be positive. @J.W.Tanner – NaturalMathLover Sep 02 '20 at 02:50
Thank you @ArturoMagidin I am new to this and just now getting the hang of it will try to use MathJAx in the future or LaTeX – NaturalMathLover Sep 02 '20 at 02:51
@NaturalMathLover: so take $s'=-s$ – J. W. Tanner Sep 02 '20 at 02:53

Prove $\exists r,s\in\Bbb{Z}$ such that $ar+ns=1\iff$ there exists an integer $b\ne 0$ such that $ab\equiv 1 \pmod n$

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