Since d divides both a and b, there are integers k and l s.t.

a = kd and b = ld.

Thus a^2 + b^2 = (k^2+l^2)d^2

and a+b = (k+l) d

Thus, the gcd of a^2+b^2 and a+b is at least d, which is a)

Now, for b), we assume that a and b are both odd.

Then k and l are odd, too.

Thus k^2+l^2 is even (odd + odd is odd and odd + odd is even)

and k+l is even.

Thus 2d divides the gcd of a^2+b^2 and a+b in this case.

Best,

ZD