Find all elements $x$ in $Z_{99}$ such that ⟨$x$⟩ $=$ ⟨$11$⟩ What I know is that ⟨$11$⟩ $=$ $(0, 11, 22, 33, 44, 55, 66, 77, 88$), but I can't understand how any elements in $Z_{99}$ can produce this result other than $11$. Any pointers?
Hint: $|x|=99/\gcd(x,99)$. Look for elements $x\in\langle 11 \rangle$ such that $|x|=9$. Then $\langle x \rangle = n\langle 11 \rangle$.
Hint: $|x|=99/\gcd(x,99)$. Look for elements $x\in\langle 11 \rangle$ such that $|x|=9$. Then $\langle x \rangle = n\langle 11 \rangle$.