prove that $(a,b)$ divides $(a+b,a-b).$

Question

In this problem, can I assume $(a,b)=1$? Otherwise I have let $d=(a+b,a-b)$. Then $d \mid ((a+b)+(a-b))$ which implies that $d \mid 2a$. Similarly, $d \mid 2b$. Therefore, $d \mid (2a,2b)$.

From here, I'm lost.

Answer 1

Let $d = (a,b)$, then we have $a=da'$ and $b=db'$, where $(a',b')=1$. Then we have: $(a+b,a-b) = (d(a'+b'),d(a'-b')) = d(a'+b',a'-b')$. Hence the proof.