For example: $7x \equiv 1 \pmod{31} $ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do?
Update
If I have a general modulo equation:
$5x + 1 \equiv 2 \pmod{6}$
What is the fastest way to solve it? My initial thought was: $5x + 1 \equiv 2 \pmod{6}$ $\Leftrightarrow 5x + 1 - 1\equiv 2 - 1 \pmod{6}$ $\Leftrightarrow 5x \equiv 1 \pmod{6}$
Then solve for the inverse of $5$ modulo 6. Is it a right approach?
Thanks,