This is the problem: In $\mathbb{Z}\_{35}$, find the inverse of $\overline{13}$ and use it to solve $\overline{13}x=\overline{9}$. 

What I've done:  

The inverse exists as gcd(35,13)=1. The euclidean algorithm gives  

35=2(13)+9  

13=1(9)+4  

9=2(4)+1  

so, eventually 1=3(35)-8(13). Thus, $8(13)\equiv$ 1 (mod 35) and $\overline{8}$ is the inverse of $\overline{13}$. BUT, the answer key says that $\overline{27}$ is the inverse and I don't see how. 

And then I'm not sure how to solve for the equation given with the said inverse.