Is it true that $x \equiv 1 \pmod{p}$? 
If $x^{m} \equiv 1 \pmod{p}$ for some prime $p$ with $\gcd(m,p-1) =1$, is it true that $x \equiv 1 \pmod{p}$?

I was wondering about this result and it seems that since the maximal order of $x$, $p-1$, is relatively prime with $m$ it might be true.

The order of $x$ divides $m$, and by Fermat's Theorem it divides $p-1$, so the order of $x$ divides the gcd of $m$ and $p-1$, which is $1$. It follows that $x\equiv 1\pmod{p}$.
