Let $a,m,s,t$ be integers, with $m > 0$ and gcd$(a,m)=1$. Show that if $a^s \equiv a^t \pmod{m}$, then $s \equiv t \pmod{\text{ord}_m(a)}$.

Question

I have been able to show this for a specific example, but I am at a loss of how to formally prove it. I know, by Euler's Theorem, that $a^{\phi(m)}\equiv 1\pmod{m}$, but I am unsure of what to do with it.

I don't need anyone to write the proof for me, but any nudges or hints in the right direction would be greatly appreciated. Thank you!