For each positive integer $n$, the set of integers $\{0,1,\ldots,n-1\}$ is known as the $\textit{residue system modulo}$ $n$. Within the residue system modulo $2^4$, let $A$ be the sum of all invertible integers modulo $2^4$ and let $B$ be the sum all of non-invertible integers modulo $2^4$. What is $A-B$?
Since $2^4$ is a power of $2$, the invertible integers are the odd ones $\{1,3,5,7,9,11,13,15\}$, and the non-invertible integers are the even ones $\{0,2,4,6,8,10,12,14\}$. Thus, \begin{align*}
A-B & = (1+3+5+7+9+11+13+15)\\
& \qquad - (0+2+4+6+8+10+12+14)\\
& = (1-0)+(3-2)+(5-4)+(7-6)+(9-8)\\
&\qquad+(11-10)+(13-12)+(15-14)\\
& = 1+1+1+1+1+1+1+1=\boxed{8}.
\end{align*}