Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.