Let $ p_i \geq 2$, $ i = 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let: \[ P = \{ x = \sum_{i = 1}^{n} x_i \prod_{j = 1, j \neq i}^{n} p_j \mid x_i \text{is a non - negative integer}, i = 1,2, \cdots n \} \] Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n - 2}{2} \cdot \prod_{i = 1}^{n} p_i$, and also find $ M$.