KöMaL 2016 A666

Let $p$ be a prime, let $k$ be a positive integer, and let $\mathcal{A}$ be a finite set of integers with at least $p^k$ elements. Denote by $N_{\text{even}}$ the number of subsets of $\mathcal{A}$ with even cardinality and sum of elements being divisible by $p^k$. Similarly, denote by $N_{\text{odd}}$ the number of subsets of $\mathcal{A}$ with odd cardinality and sum of elements being divisible by $p^k$. Show that $N_{\text{even}}\equiv N_{\text{odd}} \pmod{p}$ .