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Math Help - congruence in integer

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    congruence in integer

    if [a]=[1] in Zn, prove that (a,n)=1. Show by example that the converse may be false.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Deepu View Post
    if [a]=[1] in Zn, prove that (a,n)=1. Show by example that the converse may be false.
    I assume that [z]=\left\{m\in\mathbb{Z}:m\equiv z\text{ mod }n\right\} and \mathbb{Z}n=\mathbb{Z}_n. If, so assume that [a]=[1]\implies a\in[1]\implies a\equiv 1\text{ mod }n\implies a=zn+1\implies a+z'n=1 where z'=-z. The conclusion follows from basic knowledge about linear Diophantine equations.
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    Quote Originally Posted by Deepu View Post
    if [a]=[1] in Zn, prove that (a,n)=1. Show by example that the converse may be false.

    (3,7)=1\,\,\,but\,\,\,[3]\neq [7]\,\,\,in\,\,\,\mathbb{Z}_7

    Tonio
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