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 ${\times }_7$ 5 = Remainder obtained by dividing 4$\times$ 5 by 7
= 6

So, the composition table is as follows:

${\times }_7$	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.

Also, $3{\times }_{7}5=1$
So, 3^$-1$ = 5
$$\begin{gathered} \text{Now,} \\ {3}^{-1}{\times }_{7}4=5{\times }_{7}4=6 \end{gathered}$$