Multiplicative Inverse $ \mod p $

Question

Here's the question I'm working on:

Let $p$ be an odd prime number. Show that the multiplicative inverse of $\overline {2}$ in $\mathbb{Z_p}$ is $\overline {(p+1)/2}$. What is its multiplicative inverse if $p = 2?$

I really don't know how to even approach this problem. In order to find multiplicative inverses, I usually just compute the $gcd$ of a pair of given numbers using the Euclidian algorithm somewhere along the proof.

Answer 1

Hint $\,\ p\,$ odd $\,\Rightarrow\, p = 2k-1\,\Rightarrow\, 2k\equiv 1\pmod{\!p}$