Four positive integers $A$, $B$, $C$ and $D$ have a sum of 36. If $A+2 = B-2 = C \times 2 = D \div 2$, what is the value of the product $A \times B \times C \times D$?
We have that $A + B + C + D = 36$.  Substituting everything in terms of $C$, we find that $(2C - 2) + (2C + 2) + C + (4C) = 36$, which means that $C = 4$.  Thus $A = 6$, $B = 10$, and $D = 16$.  Therefore our desired answer is $6\cdot 10\cdot 16\cdot 4 = \boxed{3840}$.