In the group $G = \{1, 2, 3, 4, 5, 6\}$ under $\otimes_{7}$,the solution of $4 \otimes_{7} x = 5$ is

Let $^*$ be the binary operation on $N$ given by $a ^* b = \text{L.C.M. of } a \text{ and } b$. Find which elements of $N$ are invertible for the operation $^*$?

Which of the following is a subgroup of the group $G = \{2^{n} \mid n \in \mathbb{Z}\}$ under multiplication?

Which one of the following is not true?

We define a binary relation $\sim$ on the set of all $3 \times 3$ real matrices as $A \sim B$ if and only if there exist invertible matrices $P$ and $Q$ such that $B = P A Q^{-1}$. The binary relation $\sim$ is

Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^+$,define $*$ by $a * b = |a - b|$.