Some question about number theory

[henry407]

How to prove that if a-c | ab+cd then a-c | ad+cb is correct??
And how to prove the gcd(a^2+b^2, a+b) is 1 or 2. where gcd(a,b)=1.

 

[ArcanaNoir]

All I can suggest about the first part is try $(a-c)^2(ab+cd)$
That can be written as $a^3b-2a^2bc-2ac^2d+c^3d+ac(ad+cb)$
I'm sure I'm lacking some property that can get from here to the solution. Something like, if x divides y and x divides y+z then x also divides z, or similarly, if x divides y and x divides z, then x divides y+z. Properties like that, but I don't think specifically that one, come into play.

 

[ArcanaNoir]

Okay forget what I just said. Try (a-c)(b-d)