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,

\[a * e = a = e * a, \forall a \in N\] 
\[HCF\left( a, e \right) = a = HCF\left( e, a \right), \forall a \in N\] 
\[ \Rightarrow HCF\left( a, e \right) = a, \forall a \in N\] 

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