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A beautiful combination of NT and polynomials

Source: KoMaL & The Golden Digits Contest, May 2024, P3

May 19, 2024
number theorypolynomialkomal

Problem Statement

Let pp be a prime number and A\mathcal{A} be a finite set of integers, with at least pkp^k elements. Denote by NevenN_{\text{even}} the number of subsets of A\mathcal{A} with even cardinality and sum of elements divisible by pkp^k. Define NoddN_{\text{odd}} similarly. Prove that Neven≡Nodd mod p.N_{\text{even}}\equiv N_{\text{odd}}\bmod{p}.
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