Q: modular arithmetic - proof not clear I am trying to understand a proof for a presentation and I am struggeling with 1 statement. I hope this problem is not too specific.
It is one of many propositions that is used for the derivation of fermat little theorem.
The proof as image
My problem is the third line from below. In this line they want to show that there are two numbers in the same congruence class by choosing $s\geq 0$ and $0 < t < m - 1$ and they say $a^{s} ≡ a^{s+t}$.
I found a counterexample where this is not true:
Let's say $a = 3$ and $m = 7$, so a and m are coprime. Now we choose for $s = 0$ and let's see if $a^{s} ≡ a^{s+t}$. Then the left-hand side would be $3^{0} = 1$ and the right-hand side would have these values for all possible $t$: $3^{1}$,$3^{2}$,$3^{3}$,$3^{4}$,$3^{5}$. Taking these numbers mod 7 we obtain $3,2,6,4,5$ for the right-hand side. But the $1$ of the left-hand side is not congruent to any of these numbers on the right-hand-side. Maybe they made a mistake in the paper and they wrote $0 < t < m - 1$ instead of $0 < t \leq m - 1$. Or maybe I dont get the proof at all. Thanks for you help
