We prove the strong unique continuation property for many-body Schrödinger operators with an external potential and an interaction potential both in $L^p_{\rm{loc}}(\mathbb{R}^d)$, where p > 2 if d = 3 and p = max(2d/3, 2) otherwise, independently of the number of particles. With the same assumptions , we obtain the Hohenberg-Kohn theorem, which is one of the most fundamental results in Density Functional Theory.