Let $^*$ be the binary operation on $N$ given by $a \,^* \,b = \text{L.C.M. of } a \text{ and } b$. Find $5 \,^* \,7$ and $20 \,^* \,16$.

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = \frac{ab}{2}$.

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 $a * b = ab$.

Show that addition, subtraction, and multiplication are binary operations on $R$, but division is not a binary operation on $R$. Further, show that division is a binary operation on the set $R_*$ of nonzero real numbers.

If $*$ is a binary operation defined on $R$ by $a * b = 1 + ab, \forall a, b \in R$. Then the operation $*$ is:

The binary operation $*$ on $R - \{-1\}$ defined by $a * b = \frac{a}{b+1}$ is: