Let T be some PPT* with legs x and y, and hypotenuse, z. A rectangle of dimension x*y is drawn in one corner of the square on z, so that the ‘surplus’ area between the rectangle and the square is a concave hexagon of area H units.

Question 1. Prove that H is not divisible by (x+y).

Question 2. Prove that there are no (x,y) such that H is a cube.

Question 3. Prove that if H is a square, H is divisible by 13.

Alternatively in each case, provide a counter-example.

*A PPT or 'Primitive Pythagorean Triangle' is a right angled triangle with unit sides that do not have any common divisor; e.g. 3,4,5 is a PPT, but 6,8,10 is not, because the length of each side is divisible by 2.