jmc

We don't know the $y$-intercept, so let the $y$-intercept be $a$, where $1\le a\le 9$. Our line contains the points $(0,a)$ and $(4,365)$, so its slope is $\frac{365-a}{4-0}=\frac{365-a}{4}$. We are trying to find the smallest possible value of this expression, to minimize the slope. The denominator is fixed, so we must find the smallest possible value of the numerator, $365-a$. The smallest value will occur when $a$ is as large as possible, so $a=9$. Then the slope will be $\frac{365-9}{4}=\frac{356}{4}=\boxed{89}$.