GATE CS 2005 Q13 — Monoids and Groups

To find the inverse of an element $a$ in a group under multiplication modulo $15$, we need to find an element $b$ in the group such that:

Here, the identity element of the group is $1$.

1. Finding the inverse of $4$:

We test the elements of the group to find $x$ such that $4 \times x \equiv 1 \pmod{15}$:

Thus, the inverse of $4$ is $4$.

2. Finding the inverse of $7$:

We test the elements of the group to find $y$ such that $7 \times y \equiv 1 \pmod{15}$:

Thus, the inverse of $7$ is $13$.

Therefore, the inverses of $4$ and $7$ are $4$ and $13$ respectively, which corresponds to option C.