Question

Show that if a has a multiplicative inverse modulo N, then this inverse is unique (modulo N).

Short answer

It is proved that the inverse multiplicative modulo is N a distinct modulo.

Step-by-step solution

First we will find the expression of x

Initially, take the two multiplicative inverses modulo N of x as $y_1$and$y_2$ .

Then,

$$\begin{gathered} {\mathrm{xy}}_1\equiv 1(\mathrm{mod}\mathrm{N}) \\ {\mathrm{xy}}_2\equiv 1(\mathrm{mod}\mathrm{N}) \end{gathered}$$

Subtract both equations as follows:

role="math" localid="1658916438802" $$\begin{gathered} {\mathrm{xy}}_1-{\mathrm{xy}}_2\equiv 1-1(\mathrm{mod}\mathrm{N}) \\ \Rightarrow x{y}_1-{\mathrm{xy}}_2\equiv 0(\mathrm{mod}\mathrm{N}) \\ \Rightarrow x\left(y_1-y_2\right)\equiv 0(\mathrm{mod}\mathrm{N}) \end{gathered}$$

Proving modulo N as distinct modulo

Take$y_1$ as the multiplicative inverse. Then,

$$\begin{gathered} y_1·x\left(y_1-y_2\right)\equiv x^{-1}.0\left(modN\right) \\ \Rightarrow \left(y_1-y_2\right)\equiv 0\left(modN\right) \\ \Rightarrow y_1\equiv y_2\left(modN\right) \end{gathered}$$

From this, it is derived that the inverse multiplicative modulo N is a distinct modulo N .

Therefore, the inverse multiplicative modulo N is a distinct modulo.