Question

Find the inverse of 5 under multiplication modulo 11 on Z₁₁.

Answer 1

$$\begin{gathered} Z_{11}=\left\{0,1,2,3,4,5,6,7,8,9,10\right\} \\ \text{Multiplication modulo 11 is defined as follows:} \\ \text{For}a,b\in Z_{11}, \\ a{\times }_{11}b\mathrm{is}\mathrm{the}\mathrm{r}\text{emainder when}a\times b\text{is divided by 11.} \end{gathered}$$

Here,
1 ${\times }_{11}$ 1 = Remainder obtained by dividing 1 $\times$ 1 by 11
= 1

3 ${\times }_{11}$ 4 = Remainder obtained by dividing 3 $\times$ 4 by 11
= 1

4 ${\times }_{11}$ 5 = Remainder obtained by dividing 4 $\times$ 5 by 11
= 9
So, the composition table is as follows:

${\times }_{11}$	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	1	3	5	7	9
3	3	6	9	1	4	7	10	2	5	8
4	4	8	1	5	9	2	6	10	3	7
5	5	10	4	9	3	8	2	7	1	6
6	6	1	7	2	8	3	9	4	10	5
7	7	3	10	6	2	9	5	1	8	4
8	8	5	2	10	7	4	1	9	6	3
9	9	7	5	3	1	10	8	6	4	2
10	10	9	8	7	6	5	4	3	2	1

We observe that the first row of the composition table is same as the top-most row.
So, the identity element is 1.

Also,
$$\begin{gathered} 5{\times }_{11}9=1 \\ \text{Hence,}{5}^{-1}=9 \end{gathered}$$