A product decomposition of complex projective space has a cohomologically trivial factor

Suppose \(X\) and \(Y\) are spaces such that \[ X\times Y\simeq\mathbb{CP}^n, \qquad n\ge1. \] Prove that either \[ H^i(X)=0\quad\text{for all }i>0, \] or \[ H^i(Y)=0\quad\text{for all }i>0. \]