Finding the inverse of a number under a certain modulus How does one get the inverse of 7 modulo 11?
I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure.
Likewise, I have the same problem finding the inverse of 3 modulo 13, which is 9.

As said by AlexR, you can find $x,y\in\mathbb Z$ with $7x+11y=1$ using Extended Euclidean algorithm (see this answer for how to best use it).   
You can also use elementary modular arithmetic 'tricks':     
$\bmod{11}\!:\ 7x\equiv 1\equiv -21\stackrel{:7}\iff x\equiv -3\equiv 8$.   
$\bmod{11}\!:\ 7x\equiv -4x\equiv 1\equiv 12\stackrel{:(-4)}\iff x\equiv -3\equiv 8$.   
We could divide by $7$ and $-4$ because $(7,11)=(-4,11)=1$.
