I'm working on the following question:

Suppose that $a,b \in \mathbb{Z}$ and are not both $0$. Suppose also that $g = \gcd(a,b)$. Show that $d$ is a common divisor of $a$ and $b$ if and only if $d \mid g$.


Since $\gcd (a,b) = g$, we can say that $g \mid a$ and $g \mid b$. By Bezout's theorem, $\exists x,y \in \mathbb{Z}$ such that

$$ax+by = g.$$

(That's as far as I got)


I need to show that $g \mid a$ and $g \mid b$ so that $d$ can be a common divisor of $a$ and $b$.

Any suggestions as to where I should go next would be much appreciated.


$\implies$ If $d$ is a common divisor of $a$ and $b$ then $\exists i,j\in \mathbb{Z}$ s.t. $a=id$ and $b=jd$. So by Bezout's identity $\exists x,y \in \mathbb{Z}$ s.t. $$ax+by=idx+jdy=d(ix+jy)=g$$ and so $d|g$

$\Longleftarrow$ Assume $d|g$. By definition $g|b$ and $g|a$, and by transitivity of divisibility, $d$ divides both $a$ and $b$.


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