If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$
Solution 1:
Let $a$ and $b$ be coprime. Then $[a]_b$ generates $\mathbb Z/b\mathbb Z$. So there is some $x$ such that $x[a]_b=[1]_b$. By definition, this means there exists a $y$ such that $xa-1=yb$, as desired.