I am redoing exams as a preparation and I found this weird particular exercise to me.
"Does $32$ have a multiplicative inverse in modulo $77$? If yes, calculate the inverse."
Since the $\gcd(77,32)$ is $1$, it has an inverse.
However, when I calculated it using the extended euclidean algorithm, I ended up with
$1 = (-12)32 + (5)77$, which means my inverse of $32$ in mod $77$ is $-12$?
When I used an online calculator to check my answer I always got $65$, though.
I'm not quite sure I understand why or how it is $65$ and not $-12$…
I have redone my method multiple times but I always end up with $-12$
Thank you for your time in advance.
Best Answer
$$-12 \equiv 65 \pmod{77}$$
To see that notice that $65-(-12)=77$.