# If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$?

If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$?

Could anybody clear my this doubt?

Thanks for the help

Hint: Let order of element $a$ is $n$ then $order (a^m) = \dfrac {n}{(m, n)}$, where $(m, n)$ is the gcd of $m$ and $n$.
• That mean we can have $12 = \dfrac{n}{(12, n)} then possible value of n will be be the answer? – monalisa Apr 30 '13 at 18:49 • No, @monalisa. You know that $$12=\frac{n}{\gcd(5,n)}.$$ Given that$5$is a prime there aren't too many choices for$\gcd(5,n)\$. – Jyrki Lahtonen Apr 30 '13 at 19:01