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 $s \equiv t \pmod{\text{ord}_m(a)}$.

Asked Mar 20 '24 at 17:22

Active Mar 20 '24 at 21:59

Viewed 17 times

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!

edited Mar 20 '24 at 21:59

Bill Dubuque

272,048

asked Mar 20 '24 at 17:22

Zachary Czajka

See here. You have $a^{s-t}\equiv 1\bmod m$, hence $s-t\equiv 0\bmod ord_m(a)$. – Dietrich Burde Mar 20 '24 at 17:26

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 $s \equiv t \pmod{\text{ord}_m(a)}$.

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