If p is prime, prove that (a+b)^p is congruent to a^p + b^p (mod p)
If you expand (a+b)^p as a binomial the intermediatary terms $\displaystyle {p\choose k}a^pb^{p-k}$ for $\displaystyle 1\leq k\leq p-1$ become that $\displaystyle p$ divides $\displaystyle {p\choose k}$.