Given $m = 2n + 1$, what integer between 0 and $m$ is the inverse of 2 modulo $m$? Answer in terms of $n$.

$$\small{\text{
Given $m = 2n + 1$, what integer between 0 and $m$ is the inverse of
$ 2 \pmod{ m}$
}}\\
\small{\text{
Answer in terms of $n$.
}}$$

 

$$\small{\text{
$2x \equiv1 \pmod{m}$
}}\\
\small{\text{
$2x \equiv1 \pmod{2n+1}$
}}\\
\small{\text{
$2x-1 = 2n+1$
}}\\
\small{\text{
$2x = 2n+2$
}}\\
\small{\text{
$\mathbf{x = n+1}$
}}\\$$

 

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