While working to learn modern algebra (abstract algebra) in preparation for next year I stumbled across the section on multiplicative inverses. The textbook is pretty solid and I can see how the concepts from the previous sections are relating to this one. However I got stuck on Excercise 10 because I skipped Excercise 9 (I didn't understand it) ... Now I know the answer for Excercise 10 is right but I'm unable to relate "why" with my answer and I was wondering if I could get a little clarification. Below I've typed the excercises for reference... Thanks for the help

Excercise 9 (I skipped this one):

Let [a] be an element of

that has a multiplicative inverse

in

. Prove that

is the unique solution in

to the equation

Excercise 10

Solve each of the following equations by finding

and using the result in Excercise 9.

(a)

in

This format looked VERY similar to a previous excercise in which I used the Extended Eclidean Algorithm to solve so I first tried that

mod

Thus x = 11 is a solution. Now from here I checked my answer in the book and it is indeed right however they list it a separate way and I'm wondering what exactly their method says and if it's shorter. The method I've used worked for all 8 of the parts in this excercise, so the methodology is right, but I'm kind of lost as to what it means haha

Book answer:

Thanks for any help understanding what is going on here.