It is important to state under what conditions an inverse can be found.
Theorem: If a and m are relatively prime integers and m > 1, then an inverse of a modulo m exisits and is unique modulo m.
Note that inverses in general are not unique, but there is one unique inverse modulo m. In the example above, any member of [−4]19 is an inverse of 14 modulo 19; the one unique inverse modulo 19 is 15.