Question:
Given a right triangle whose side lengths are all integer multiples of 8, how many units are in the smallest possible perimeter of such a triangle?

Answer:
The smallest right triangle with integer lengths is the $3 - 4 - 5$ right triangle.  Since none of those lengths are multiples of 8, we must scale each side up by a factor of 8 to create a right triangle whose side lengths are all integer multiples of 8.  This triangle has perimeter $3\cdot 8 + 4 \cdot 8 + 5 \cdot 8 = (3 + 4 + 5 )\cdot 8 = 12\cdot 8 = \boxed{96}$ units.