I have two questions.
When computing the modular inverse of a number (computing $x^{-1} \bmod m$), in practice do people actually do "Mentally, what multiple of $x$, taken mod $m$, has a remainder of $1$?"
I don't understand the Extended Euclidean Algorithm. I know how to get $\gcd(a,b)$ using the normal Euclidean Algorithm and how to write the modular equations out in the form $a = qp + r$. For example $\gcd(\color{blue}{22},\color{red}{6})$ implies an equation $\color{blue}{22} = (3)(\color{red}{6}) + \color{green}{4}$ which leads to $\gcd(\color{red}{6},\color{green}{4})$. But I don't understand how you "back up" from the final step (when you compute the final gcd once $b=0$) to get the original terms $x,y$ in the first modular equation.
I hope this makes sense. Thanks.