To prove that if a-c divides ab+cd, then a-c also divides ad+cb, one approach involves manipulating the expressions and applying properties of divisibility. The suggestion to use (a-c)^2(ab+cd) leads to a complex polynomial that may not directly simplify the proof. Instead, exploring the relationship through (a-c)(b-d) could provide a clearer path to the solution. For the second question regarding gcd(a^2+b^2, a+b), it is established that the gcd can only be 1 or 2 when gcd(a,b)=1, based on the properties of coprime integers. These discussions highlight the importance of leveraging divisibility properties and gcd characteristics in number theory.
