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Generating Z^2

https://matholympiad.org.bd/forum/viewtopic.php?f=27&t=3297

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Generating Z^2

Posted: Sat Feb 21, 2015 1:13 am
by nayel
Let $a=(a_1,a_2)\in\mathbb Z^2$ where $a_1$ and $a_2$ are coprime. Show that there exists $b=(b_1,b_2)\in\mathbb Z^2$ such that any element of $\mathbb Z^2$ can be written as a $\mathbb Z$-linear combination of $a$ and $b$.

Terminology:
1. A $\mathbb Z$-linear combination of $a$ and $b$ is anything of the form $xa+yb=x(a_1,a_2)+y(b_1,b_2):=(xa_1+yb_1,xa_2+yb_2)$ for $x,y\in\mathbb Z$.
2. We also say $a$ and $b$ generate $\mathbb Z^2$ as a $\mathbb Z$-module.
Follow-up question:
(Generalize to $\mathbb Z^n$) Given $e_1\in\mathbb Z^n$ such that its coordinates are coprime, can we find $e_2,\dots,e_n\in\mathbb Z^n$ such that they generate $\mathbb Z^n$ (as a $\mathbb Z$-module)?
Remark:
If the coordinates aren't coprime then none of the above is possible. (Why?)
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