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Find $ord_m b^2$ if $ord_m a = 10$ and $ab\equiv 1\pmod m$

If $ab \equiv 1 \pmod{m}$ and if $ord_ma=10$, find $ord_mb^{2}$.

I know that $ab \equiv 1 \pmod{m}$ is used to find multiplicative inverses, and I know the basics to orders of elements, but I'm not sure how I would go about combining what I know to solve this.

Thanks

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1 Answers 1

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Working modulo $\,m\,$:

$ab=1\,\,,\,\,a^{10}=1\Longrightarrow 1=(ab)^{10}=a^{10}\cdot b^{10}=b^{10}=\left(b^2\right)^5$

End now the argument.