Question

For the binary operation ×₇ on the set S = {1, 2, 3, 4, 5, 6}, compute 3⁻¹ ×₇ 4.

Answer 1

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,

\[ 3^{- 1} \times_7 4 = 5 \times_7 4 = 6\]