note that a^2+b^2 = |z1|^2, and c^2+d^2 = |z2|^2 (this is the complex modulus, if z1 = a+ib, then |z1| = √(a^2 + b^2)).

in the complex numbers, is it true that |(z1)(z2)| = |z1||z2|?

if so (this is really what you must prove), then |(z1)(z2)|^2 = (|z1||z2|)^2 = |z1|^2|z2|^2, so we can take u and v to be the real and imaginary parts of z1z2.

after that, it is not hard to see that u and v must be integers.