 Solution

Multiplication, followed by induction and inversion for negative powers, givesIf $c\ne 0$, apply Euclidean division to write $a=q|c|+r$ with $0\le r<|c|$. Choose $n=-q\mathrm{sgn}(c)$, so $a+nc=r$. This is one step of the Euclidean algorithm implemented by a determinant-one row operation.