For every $a \in \mathbb{N}$, denote by $M(a)$ the number of elements in the set
\[ \{ b \in \mathbb{N} \mid a + b \text{ is a divisor of } ab \}. \]
Find $\max_{a \leq 1983} M(a).$

Let's think step by step and output the final answer within \boxed{}.
user