Euler's theorem and $Z_p^*$

Question

Euler's theorem states that for x and p coprime, $x^{\phi(p)}\equiv1 \bmod p$, does it mean that $x$ has the order $\phi(p)$ in $Z_p^*$? I checked it manually for $x=2$ and it held for some cases. How would one prove or disprove that?

Answer 1

This theorem only says that $\text{ord}_p(x)$ must divides $\phi(p)$.

For the counter-example let $p=7$ and let $x=2$;
but notice that $\text{ord}_7(2)= 3$. $$ 2^3 \overset{7}{\equiv}1 \ \ \text{and} \ \ 2^2 \overset{7}{\ncong}1 \ \ \text{and} \ \ 2^1 \overset{7}{\ncong}1 .$$