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Find the Inverse of 5 Under Multiplication Modulo 11 on Z11. - Mathematics

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Question

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

Sum
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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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Chapter 3: Binary Operations - Exercise 3.5 [Page 33]

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RD Sharma Mathematics [English] Class 12
Chapter 3 Binary Operations
Exercise 3.5 | Q 7 | Page 33

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