(note then that it is true that if both sides have a common factor that divides the module we can divide everything - both sides and the module - by that number )
In fact so you have ...
More generally, if you wanted to find the modular inverse here are a couple of ways:
1. Use the extended euclidean algorithm -this works always.
2. If the module were a prime , note that by Fermat's Little Theorem, since then and are modular inverses, and so you compute by repeatedly squaring and taking modules at each step.