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?

Let e be the identity element. Then,

a * e=a=e * a, aNHCF a, e=a=HCF e, a, aNHCF a, e=a, aN

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

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