2. Given the equation in terms of $x$
$$
\frac{5}{2} x-a=\frac{8}{5} x+142 \text {, }
$$

when $a$ is some positive integer, the solution of the equation is a positive integer. Then the smallest positive integer value of $a$ is ( ).
(A) 2
(B) 3
(C) 4
(D) 5
2. A.

From the original equation, we get $a=\frac{9}{10} x-142$.
Since $a$ is a positive integer, we have,
$$
\frac{9}{10} x>142 \Rightarrow x>157 \frac{7}{9} \text {. }
$$

Also, $x$ is a positive integer, to make $\frac{9}{10} x$ an integer, $x$ must be a multiple of 10, and to make $a$ as small as possible, we should take $x=160$. Therefore, $a=\frac{9}{10} \times 160-142=2$.
Thus, the smallest positive integer $a$ that satisfies the condition is 2.