Question

The set $\{1,2,4,7,8,11,13,14\}$ is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively

• A. 3 and 13
• B. 2 and 11
• C. 4 and 13
• D. 8 and 14

Answer 1

The correct option is C 4 and 13

The set $S=\{1,2,4,7,8,11,12,14\}$ is a group under multiplication modulo 15.
The identity element for this group is $e=1\sin \alpha \forall xϵS,1.x\text{mod}15=x$
Now let the inverse of 4 be $4^{-1}$
Now $\left({4.4}^{-1}\right)\text{mod 15 =e=1}$
Since $\left(4.4\right)\text{mod 15=1}$
$\therefore 4^{-1}=4$ (This inverse is unique)
Similarly let the inverse of 7 be $7^{-1}$
$\left({7.7}^{-1}\right)\text{mod}15=1$
putting each element of set as $7^{-1}$ by trial and error we get
$\left(7.13\right)\text{mod 15 = 91 mod 15 =1}$
$\therefore 7^{-1}=13$
So, $4^{-1}$ and $7^{-1}$ are respectively 4 and 13.
Correct choice is (c)