Answer: C

Small Hint:

Let the intercepts be $p$ and $q$, and use $\frac{4}{p}+\frac{3}{q}=1$

Big Hint:

Rearrange to $q=3+\frac{12}{p-4}$

Solution:

Let the $x$-intercept be the prime $p$ and the $y$-intercept be the positive integer $q.$ Intercept form gives $$\begin{gathered} \frac4p+\frac3q=1, \\ q=\frac{3p}{p-4}=3+\frac{12}{p-4}. \end{gathered}$$ Thus $p-4$ is a positive divisor of $12.$ Testing $1,2,3,4,6,12$ gives prime $p$ only for $p=5$ and $p=7.$ Therefore there are $2$ lines. Thus the correct answer is C.