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Question
For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.
Solution
Finding identity element:
Here,
1 \[\times_7\] 1 = Remainder obtained by dividing 1\[\times\] 1 by 7
= 1
3 \[\times_7\] 4 = Remainder obtained by dividing 3 \[\times\] 4 by 7
= 5
4 \[\times_7\] = Remainder obtained by dividing 4 \[\times\] 5 by 7
= 6
So, the composition table is as follows:
| ×7 | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 2 | 4 | 6 | 1 | 3 | 5 |
| 3 | 3 | 6 | 2 | 5 | 1 | 4 |
| 4 | 4 | 1 | 5 | 2 | 6 | 3 |
| 5 | 5 | 3 | 1 | 6 | 4 | 2 |
| 6 | 6 | 5 | 4 | 3 | 2 | 1 |
We observe that all the elements of the first row of the composition table are same as the top-most row.
So, the identity element is 1.
\[3 \times_7 5 = 1\]= 5
So, 3-1 = 5
Now,
\[ 3^{- 1} \times_7 4 = 5 \times_7 4 = 6\]
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