The inverse of $2010$ in the group $Q^{+}$ of all positive rational numbers under the binary operation $*$ defined by $a * b = \frac{ab}{2010}, \forall a, b \in Q^{+}$,is

Let $^*$ be the binary operation on $N$ defined by $a \,^*\, b = \text{H.C.F. of } a \text{ and } b$. Is $^*$ commutative? Is $^*$ associative? Does there exist an identity for this binary operation on $N$?

On the set of all non-zero reals,an operation $*$ is defined as $a * b = \frac{3ab}{2}$. In this group,a solution of $(2 * x) * 3^{-1} = 4^{-1}$ is

Show that $0$ is the identity for addition on $R$ and $1$ is the identity for multiplication on $R$. But there is no identity element for the operations $-: R \times R \rightarrow R$ and $\div : R_* \times R_* \rightarrow R_*$.

Consider the binary operations $^*: R \times R \rightarrow R$ and $o: R \times R \rightarrow R$ defined as $a \,^*\, b = |a-b|$ and $a \,o\, b = a$,$\forall \, a, b \in R$. Show that $^*$ is commutative but not associative,and $o$ is associative but not commutative. Further,show that $\forall \, a, b, c \in R, a \,^*\, (b \,o\, c) = (a \,^*\, b) \,o\, (a \,^*\, c)$. [If it is so,we say that the operation $^*$ distributes over the operation $o$]. Does $o$ distribute over $^*$? Justify your answer.

Show that addition and multiplication are associative binary operations on $R$. However,subtraction is not associative on $R$,and division is not associative on $R_*$.