Proof verification: $\gcd(a,n)=1$ iff $ab\equiv 1\pmod{n}$ for some integer $b$.
Can someone please verify whether my proof is logically correct? :)
Proof:
Let $\gcd(a,n)=1$. Then there exist integers $r$ and $s$ such that $ar+ns=1$. Then $ns = 1 - ar$. Then $ar\equiv 1\pmod n$ by definition. If $[b]_{n}=[r]_{n}$, then $ab\equiv 1\pmod n$.
Let $ab\equiv 1$ for some integer $b$. Then $n$ divides $ab-1$. Then there exists an integer $k$ such that $ab-1=nk$ or $ab-nk=1$. Let $d=\gcd(a,n)$. Since $d\mid a$ and $d\mid n$, then $d\mid ab-nk$. Then $d\mid 1$. For that to be true, $d=1$. Therefore, $\gcd(a,n)=1$. $\square$