Prime Power of Sum Modulo Prime

Theorem
Let $$p$$ be a prime number.

Then $$\left({a + b}\right)^p \equiv a^p + b^p \pmod p$$.

Proof
From the Binomial Theorem: $$\left({a + b}\right)^p = \sum_{k=0}^p \binom p k a^k b^{p-k}$$.

Also note that $$\sum_{k=0}^p \binom p k a^k b^{p-k} = a^p + \sum_{k=1}^{p-1} \binom p k a^k b^{p-k} + b^p$$.

So:

$$ $$ $$ $$

Also see
Compare with the Freshman's Dream.