Order of the group containing $a^m$ and $a^n$

Question

Given that $G=\langle a\rangle$, I have to find the order of $\langle a^m,a^n\rangle$. Now I know that $\langle a^m, a^n\rangle = \langle a^{\gcd(m,n)}\rangle$. Also my book says that the order of this group is $lcm(|a^m|,|a^n|)$. I don't know how they got this. The way I tried was : order of the group = order of the generator = $|a|/\gcd[|a|,\gcd(m,n)]$. Any suggestions?