Bezout lemma unique solution

Question

If i take $a$ and $b$ such as $GCD(a,b)=1$ then for the bezout lemma the inverse of $a$ mod $b$ will exist . If $GCD(a,b)\neq 1$ there can be multiple inverses of $a$ mod $b$ or simply the inverse doesn't exist ?

Answer 1

The inverse doesn't exist. If $(a, b) = m$, then $m$ will divide any linear combination of $a$ and $b$. For $m\neq 1$ no multiple of $a$ will be equivalent to $1$ mod $b$.