jmc

An integer congruent to $5(\mathrm{mod}13)$ can be written as $13n+5$. Therefore, we have the inequality $$-200\le 13n+5\le 200.$$We solve for the inequality by subtracting each term by $5$ and then dividing by $13$ to get $$-205\le 13n\le 195⟹-\frac{205}{13}\le n\le \frac{195}{13}.$$The smallest integer greater than $-\frac{205}{13}$ is $-15$ and the largest integer less than $\frac{195}{13}$ is $15$. There are $\boxed{31}$ integers from $-15$ to $15$ inclusive.