Question

Let * be the binary operation on N defined by
a * b = HCF of a and b.
Does there exist identity for this binary operation one N?

Answer 1

Let e be the identity element. Then,

$$\begin{gathered} a*e=a=e*a,\forall a\in N \\ \text{HCF}\left(a,e\right)=a=\text{HCF}\left(e,a\right),\forall a\in N \\ \Rightarrow \text{HCF}\left(a,e\right)=a,\forall a\in N \end{gathered}$$

We cannot find e that satisfies this condition.
So, the identity element with respect to * does not exist in N.