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

## StatementEdit

If *p* is a rational prime, for all integers *a* ≠ 0,

## ProofsEdit

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

- .