If $G=\langle x\rangle$ is cyclic group and order of $G$ is $40$ then how many order of $x^3$

Question

If $G=\langle x\rangle$ is cyclic group and order of $G$ is $40$ then how many order of $x^3$

Answer 1

$3$ is relatively prime to $40$, so there is an integer $k$ (in this case $k = 27$) such that $3k \equiv 1 \pmod {40}$. Thus $\langle x\rangle \supset \langle x^3\rangle \supset \langle(x^{27})^{3}\rangle = \langle x\rangle$, so $x^3$ is also order $40$.