Question
For the binary operation $\times _7$ on the set $S =\{1, 2, 3, 4, 5, 6\},$ compute $3^{−1} \times _7 4.$

Answer

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,  $3 ^{- 1 } = 5$ 
Now,
$ 3 ^{- 1 }\times _7  4  =  5 \times _7  4  =  6$ 

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