Question

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

Answer 1

\[Z_{11} = \left\{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right\}\]
\[\text{Multiplication modulo 11 is defined as follows}:\]
\[\text{For a, b }\in Z_{11} , \]
\[a \times_{11} \text{b is the remainder when }a\times b \text { is divided by }11.\]

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:

×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,

 \[5 \times_{11} 9 = 1\]
\[\text{Hence}, 5^{- 1} = 9\]