If $a * b = 10ab$ on $Q^{+}$,then find the inverse of $0.01$.

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

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.

The set $\{-1, 0, 1\}$ is not a multiplicative group because of the failure of

Show that subtraction and division are not binary operations on the set of natural numbers $N$.

State whether the following statement is true or false and justify your answer: If $^*$ is a commutative binary operation on $N$,then $a ^* (b ^* c) = (c ^* b) ^* a$.