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Question

For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.

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Solution

Here,

1 ×10 1 = Remainder obtained by dividing 1 × 1 by 10
= 1

3 ×10 1 = Remainder obtained by dividing 3 × 1 by 10
= 3

7 ×10 3 = Remainder obtained by dividing 7 × 3 by 10
= 1

3 ×10 3 = Remainder obtained by dividing 3× 3 by 10
= 9

So, the composition table is as follows:
×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 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.
a * 1=1 * a=a, aS
So, the identity element is 1.

Also,
3 ×10 7 = 1
3-1 = 7

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