Let u and v be two positive integers and u > v.

(a). Prove that (a, b, c) = (u2 v2, 2uv, u2 + v2) is a Pythagorean triple.

(b). Give an example that gcd(u, v) = 1 and (u^2 v^2,2uv,u^2 +v^2) is not a primitive Pythagorean triple.

(c). As we have seen in Part (b), the coprime condition is not enough to ensure that (u^2 v^2,2uv,u^2 +v^2). Give an extra condition on u and v such that (u^2 v^2,2uv,u^2 +v^2) is a primitive Pythagorean triple. Justify your answer.