Proof Using The Multinomial Expansion
The proof is a very simple application of the Multinomial formula which is brought here for the sake of simplicity.
The summation is taken over all sequences of nonnegative integer indices k1 through km such the sum of all ki is n.
Thus if we express a as a sum of 1s (ones), we obtain
Clearly, if p is prime, and if kj not equal to p for any j, we have
and
if kj equal to p for some j
Since there are exactly a elements such that the theorem follows.
Read more about this topic: Proofs Of Fermat's Little Theorem
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