I was hoping someone could help look at my attempt at this proof and how I might finish it.
By Bezout's identity, can rewrite our given information as... $\gcd(a,b)=d \Longrightarrow ax+by=d$
Dividing this equation by $d$, we get $\frac{a}{d}x+\frac{b}{d}y=1$
Bezout's Identity states that if $\frac{a}{d}, \frac{b}{d} \in \mathbb Z^+$, then there exists $x,y$ such that $\frac{a}{d}x+\frac{b}{d}y=\gcd(\frac{a}{d},\frac{b}{d})$.
Since we have already concluded that $\frac{a}{d}x+\frac{b}{d}y=1$, $\gcd(\frac{a}{d},\frac{b}{d})=1$.
Is this a complete proof? I think I have to show that $\frac{a}{d}$ and $\frac{b}{d}$ are in fact integers, but not sure how I do that.
Thank you!!