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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.

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    It is asking about nonexistent object.. – 2017-02-27
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    Check that $\overline{(p+1)/2}$ satisfies the requirements of being a multiplicative inverse of $\overline{2}$ in $\Bbb{Z}_p$. That's all there is to it! About the second question: If $p=2$ then $\overline{2}=\overline{0}$. Have you not been taught what they say about division by zero? – 2017-02-27
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    I know that but *how* should I begin is the real question. – 2017-02-27
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    Just calculate. If you are asked to show that the last digit of $3\cdot7$ is $1$ you just calculate the product, right? It's the same thing here. You calculate whatever product you are required to calculate. – 2017-02-27
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    Correct me if I'm wrong but I just calculate the product of $2$ and $(p+1)/2$? – 2017-02-27
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    You get theoretical backing from the fact that the inverse is unique. So if you find one element that works you can stop looking (and your teacher has no basis to complain). – 2017-02-27
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    Yes, that is basically it. You also need to check the result modulo $p$ because in this universe two numbers are called equal whenever their difference is multiple of $p$. – 2017-02-27

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