Question

For the binary operation ×₁₀ on set S = {1, 3, 7, 9}, find the inverse of 3.

Answer 1

Here,

1 ${\times }_{10}$ 1 = Remainder obtained by dividing 1 $\times$ 1 by 10
=1
3 ${\times }_{10}$ 7 = Remainder obtained by dividing 3 $\times$ 7 by 10
=1
7 ${\times }_{10}$ 9 = Remainder obtained by dividing 7 $\times$ 9 by 10
= 3

So, the composition table is as follows:

${\times }_{10}$	1	3	7	9
1	1	3	7	9
3	3	9	1	7
7	7	1	9	3
9	9	7	3	1

We observe that the elements of the first row are same as the top-most row.
So, $1\in S$ is the identity element with respect to ${\times }_{10}$.

Finding inverse of 3:

From the above table we observe,
3 ${\times }_{10}$ 7 = 1

So, the inverse of 3 is 7.