IF $a^m=b^n=1$ show $c^k=1$, $k$ least common multiple of m and n

Question

I am currently in a linear algebra course and the professor just lectured over finite fields. I haven't seen much of them so I am lost on a problem.

Let $a$ and $b$ be nonzero elements of a finite field $\mathbb F$, and let $m$ and $n$ be positive integers satisfying $a^m=b^n=1$. show that there exist a nonzero element $c$ of $\mathbb F$ satisfying $c^k=1$, where $k$ is the least common multiple of $m$ and $n$.

Any insight on the problem would be helpful.

Answer 1

Hint: why not try $c = ab{}{}$?