Intereting Posts

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If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax-by=1$.
I don't understand why the inverse is this?
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Integral $\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx$

I have this question:

Let $x, n$ be integers with $n \geq 2$ and $n$ not dividing $x$. Show that the order o($\bar{x}$) of $x \in Z_n$ is

$o(\bar{x})= \frac{n}{HCF(x, n)}$

I’ve been thinking about it for ages but I still don’t get why. A hint would be appreciated.

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**Hint:** $\gcd(x,n) \cdot \operatorname{lcm}(x,n)=nx$.

**Hint** $\ $ Denote $\rm\:gcd(x,y)\:$ by $\rm\:(x,y),\:$ and $\rm\:lcm(x,y)\:$ by $\rm\:[x,y].\ $ Below are two proofs.

$(1)\ \ \begin{eqnarray}\rm kx\equiv 0\iff n\mid kx \iff n\mid kx,kn\iff n\mid (kx,kn)=k\, (x,n)\iff n/(x,n)\mid k

\end{eqnarray}$

$(2)\ \ \begin{eqnarray}\rm kx\equiv 0\iff n\mid kx \iff x,n\mid kx\iff [x,n]\mid kx \iff [x,n]/x\mid k\iff n/(x,n)\mid k\end{eqnarray}$

where the last $\iff$ employs the fundamental $\rm\:lcm * gcd\:$ law: $\rm\: [x,n](x,n)\, =\, xn$.

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