Euler's (Ivory's?) proof of Fermat's little theorem.

This proof relies on the simple, but powerful observation that for prime p , and integer (mod p ), one has

, ... , ... , (mod p )
in some order

A small prime numerical demonstration:

> restart;

> p := 23;

> a := 12;

> seq(a*k mod p, k=1..p-1);

> sort([seq(a*k mod p, k=1..p-1)]);

>

Then (mod p ), from which it follows that (mod p ).