Proofs of Fermat's Little Theorem - Simplifications

Simplifications

Some of the proofs of Fermat's little theorem given below depend on two simplifications.

The first is that we may assume that a is in the range 0 ≤ a ≤ p − 1. This is a simple consequence of the laws of modular arithmetic; we are simply saying that we may first reduce a modulo p.

Secondly, it suffices to prove that

for a in the range 1 ≤ a ≤ p − 1. Indeed, if (X) holds for such a, multiplying both sides by a yields the original form of the theorem,

On the other hand, if a equals zero, the theorem holds trivially.