Prove that the area of a Pythagoras triple is divisible by 2, 3 and then 6

Question

Prove that the area of a Pythagoras triple is divisible by 2, 3 and then 6. I know that in the Pythagoras triple (x,y,z) area is equal to 1/2(xy). But I am unsure how to prove this statement.

Answer 1

You can use Euclid's formula for Pythagorean triangles. Every Pythagorean triangle is a multiple of a primitive triangle, were the legs have no common factor. The primitive triangles have $a=m^2-n^2, b=2mn, c=m^2+n^2$ with $m,n$ coprime and of opposite parity. As they are of opposite parity one is even, so $b$ is a multiple of $4$ and the area is even. If neither $m$ nor $n$ is a multiple of $3$ then $a$ is, if one is a multiple of $3$ then $b$ is. Either way the area is a multiple of $3$.