Show that if p is a prime, then (a + b)^p ≡ a^p + b^p (mod p)

(Hint: prove and use that p choose k is a multiple of p for every k ∈ {1, . . . , p − 1}). Then expand (a + b)^p via the binomial theorem).

Using the previous formula, prove Fermat’s little Theorem (in its equivalent form, a^p ≡ a (mod p)) by induction on a.

Any help appreciated!