The mean of four distinct positive integers is 5. If the largest of these four integers is 13, what is the smallest of the four integers?
Let the four distinct positive integers be $a$, $b$, $c$, and 13, with $a<b<c<13$. We have the equation $\frac{a+b+c+13}{4}=5\Rightarrow a+b+c=7$. Since $a$, $b$, and $c$ must be positive and distinct, $a=1$, $b=2$, and $c=4$. So the smallest integer is $\boxed{1}$.