I want to check the validity of my proof to the following question.
Prove that (a,b) = (a,b,a+b) and more generally (a,b) = (a,b,ax+by).
Note that (a,b) is the greatest common divisor of $a$ and $b$, $a$ and $b$ are integers.
let $(a,b)$ = $g$ and $(a,b,a+b)$ = $g^*$. Since $g$|$a$, $g|b$ $\rightarrow$ $g|a+b$ implying that $g|g^*$. On the other hand, $g^*|a$, $g^*|b$, following that $g^*|g$.
From the previous argument, $g=\pm g^*$. However, $g,g^*$, by definition, cannot be negative, thereby, $g=g^*$ as required. The general case can be proved analogously.
Thanks in advance