Famous Theorems of Mathematics/Number Theory/Fermat's Little Theorem

Statement If p is a rational prime, for all integers a ≠ 0, Proofs There are many proofs of Fermat's Little Theorem. Proof 1 (Bijection) Define a function (mod p). Let S={1,2,...,p-1} and T=f(S)={a,2a,...,(p-1)a}. We claim that these two sets are identical mod p. Since all integers not equal to 0 have inverses mod p, for any integer m with 1≤m<p, . Then is surjective. In addition, if , then and . Then is injective, and is bijective between S and T. Then, mod p, the product of all of the elements of S will be equal to the product of elements of T, meaning that and that . Then .