Fermat's little theorem as a consequence of $(x+y)^p=x^p+y^p$ in $\Bbb Z_p$?

Question

Exercise in a book I am currently reading asks that I prove Fermat's little theorem as a consequence of the fact that in $\Bbb Z_p$ for any $x$ and $y$, $(x+y)^p=x^p+y^p$. Although this feels like it should be something very simple, the solution escapes me.

Edit: rephrased the question to avoid ambiguity

Answer 1

Here's one way to see it : this result implies that the set of $x$ such that $x^p=x$ is actually a subfield of any characteristic $p$ field $K$.

Taking $K=\mathbb{Z/pZ}$, we see that it doesn't have many subfields...

Answer 2

Hint: Use induction to show that $x^p = x$. For the inductive step, show that $x^p = x$ implies that $(x+1)^p = (x+1)$.