1
Question
Write the inverse of 5 under multiplication modulo 11.
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Solution
Here,
3 4= Remainder obtained by dividing 3 x 4 by 11
= 0
4 5 = Remainder obtained by dividing 4 x 5 by 11
= 9
So,
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 coincides with the top-most row and the first column coincides with the left-most column.
These two intersect at 1.
So, the identity element is 1.
Also,
3 4= Remainder obtained by dividing 3 x 4 by 11
= 0
4 5 = Remainder obtained by dividing 4 x 5 by 11
= 9
So,
|
|
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 coincides with the top-most row and the first column coincides with the left-most column.
These two intersect at 1.
So, the identity element is 1.
Also,
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