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