Question

For the binary operation multiplication modulo 5 (×₅) defined on the set S = {1, 2, 3, 4}. Write the value of \[\left( 3 \times_5 4^{- 1} \right)^{- 1}.\] 

Answer 1

Here, 

1 \[\times_5\] 1 = Remainder obtained by dividing 1 × 1 by 5
          = 1

3 \[\times_5\] 4 = Remainder obtained by dividing 3 \[\times\] 4 by 5
           = 2

4 \[\times_5\] 4 = Remainder obtained by dividing 4 \[\times\] 4 by 5
            = 1 

So, the composition table is as follows :

×5	1	2	3	4
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

We observe that the first row of the composition table coincides with the top-most row and the first column coincides with the left-most column.

These two intersect at 1.

\[\Rightarrow a \times_5 1 = 1 \times_5 a = a, \forall a \in S\]

Thus, 1 is the identity element.

\[\text{ Now },\]
\[ \left( 3 \times_5 4^{- 1} \right)^{- 1} \]
      \[ = \left( 3 \times_5 4 \right)^{- 1 } \left [ \because 4 \times_5 4 = 1 \right]\]
       \[ = 2^{- 1} \]
       \[ = 3 \left[ \because 2 \times_5 3 = 1 \right]\]