3. Given a linear function $y=a x+b$ (where $a$ is an integer) whose graph passes through the point $(98,19)$, its intersection with the $x$-axis is $(p, 0)$, and its intersection with the $y$-axis is $(0, q)$. If $p$ is a prime number and $q$ is a positive integer, then the number of all linear functions that satisfy the conditions is ( ).
(A) 0
(B) 1
(C) 2
(D) An integer greater than 2
3. (A).

From the problem, we have $\left\{\begin{array}{l}98 a+b=19, \\ p=-\frac{b}{a}, \\ q=b .\end{array}\right.$
Since $q$ is a positive integer, by (3) we know that $b$ is a positive integer. Therefore, from (2) we know that $a$ is a negative integer.
From (i) we get $\dot{i}=19-98 a$, substituting into (2) we get
$t=-\frac{19-98 a}{a}-98-\frac{19}{a}$.
Since $19=(-1) \times(-19)$,
Therefore, when $a=-1$, $p=117$, which is not a prime number;
When $a=-19$, $p=99$, which is also not a prime number.
Thus, there does not exist a linear function that satisfies the conditions.