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Question
Find the inverse of 5 under multiplication modulo 11 on Z11.
Solution
\[Z_{11} = \left\{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right\}\]
\[\text{Multiplication modulo 11 is defined as follows}:\]
\[\text{For a, b }\in Z_{11} , \]
\[a \times_{11} \text{b is the remainder when }a\times b \text { is divided by }11.\]
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,
\[5 \times_{11} 9 = 1\]
\[\text{Hence}, 5^{- 1} = 9\]
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