smc

The difference in the $y$-coordinates is $281-17=264$, and the difference in the $x$-coordinates is $48-3=45$. The gcd of 264 and 45 is 3, so the line segment joining $(3,17)$ and $(48,281)$ has slope $$\frac{88}{15}.$$ The points on the line have coordinates $$\left(3+t,17+\frac{88}{15}t\right).$$ If $t$ is an integer, the $y$-coordinate of this point is an integer if and only if $t$ is a multiple of 15. The points where $t$ is a multiple of 15 on the segment $3\le x\le 48$ are $3$, $3+15$, $3+30$, and $3+45$. There are 4 lattice points on this line. Hence the answer is $\boxed{4}$.