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?

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?

Does this mean the side lengths and not the hypotenuse or does it mean all three sides?

Answer 1

We have $$(8a)^2+(8b)^2=(8c)^2$$ Dividing by 64: $$a^2+b^2=c^2$$ As is well-known, the smallest possible integer values for these variables are $a=3,b=4,c=5$, so the smallest possible perimeter is $$8(a+b+c)=8\cdot12=96$$

Answer 2

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.