2. Given positive integers $a, b (a>b)$ such that the sum of $a+b$, $a-b$, $a b$, and $\frac{a}{b}$ is 243. Then the number of pairs $(a, b)$ with this property ( ).
(A) has exactly 1 pair
(B) has exactly 2 pairs
(C) has a finite number of pairs, but more than 1 pair
(D) does not exist