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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
Find the inverse of $5$ under multiplication modulo $11$ on $Z_{11}.$
✓

Answer

$Z_{11} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
Multiplication modulo $11$ is defined as follows:
For $\text{a},\text{b}\in\text{Z}_{11}$,
$a\times _{11}b$ is the remainder when $a \times b$ is divided by $11.$
Here,
$1\times _{11}1 = $ Remainder obtained by dividing $1 \times 1$ by $11 = 1$
$3\times _{11}4 =$ Remainder obtained by dividing $3 \times 4$ by $11 = 1$
$4\times _{11}5 =$ Remainder obtained by dividing $4 \times 5$ by $11 = 9$
$\times _{11}$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$1$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$2$ $2$ $4$ $6$ $8$ $10$ $1$ $3$ $5$ $7$ $9$
$3$ $3$ $6$ $9$ $1$ $4$ $7$ $10$ $2$ $5$ $8$
$4$ $4$ $8$ $1$ $5$ $9$ $2$ $6$ $10$ $3$ $7$
$5$ $5$ $10$ $4$ $9$ $3$ $8$ $2$ $7$ $1$ $6$
$6$ $6$ $1$ $7$ $2$ $8$ $3$ $9$ $4$ $10$ $5$
$7$ $7$ $3$ $10$ $6$ $2$ $9$ $5$ $1$ $8$ $4$
$8$ $8$ $5$ $2$ $10$ $7$ $4$ $1$ $9$ $6$ $3$
$9$ $9$ $7$ $5$ $3$ $1$ $10$ $8$ $6$ $4$ $2$
$10$ $10$ $9$ $8$ $7$ $6$ $5$ $4$ $3$ $2$ $1$
We observe that the first row of the composition table is same as the top-most row.
Therefore,
The identity element is $1.$
Also,
$5\times _{11}9 = 1$
Hence, $5 - 1 = 9$

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