jmc

The given information can be expressed by the congruences $$\begin{array}{rl}n+5\equiv 0(\mathrm{mod}8)⟹& n\equiv 3(\mathrm{mod}8),\\ n-8\equiv 0(\mathrm{mod}5)⟹& n\equiv 3(\mathrm{mod}5).\end{array}$$Since $\gcd (5,8)=1$, the above congruences imply $n\equiv 3(\mathrm{mod}40)$. This also implies the original congruences so $n$ is of the form $3+40m$ for an integer $m$. The first few positive integers of that form are $3,43,83,123$. Thus $\boxed{123}$ is the smallest such number with three digits.