Number theory Higher algebra by Bernard and Child

Question

If a and b are prime to each other, then show that

-) a+b and a - b have no common factor other than 2

-) a^2- ab + b^2 and a+b have no common factor other than 3

Answer 1

If prime $d(>1)$ divides both $a+b, a^2-ab+b^2$

$d$ must divide $(a+b)^2-(a^2-ab+b^2)=3ab$

If $d$ divides $a,d$ must divide $a+b-a=b\implies d\mid(a,b)=1$

Similarly, $d$ cannot divide $b$

Answer 2

Hint:

$\gcd(a+b,a-b) \mid (a+b)+(a-b)$