Question

Let $\ast$ be the binary operation on N defined by $a\ast b=HCF$ of a and b. Is $\ast$ commutative? Is $\ast$ associative? Does there exist identity for this binary operation on N?

Answer 1

The bianry operation $\ast$ on N is defined as $a\ast b=HCF$ of a and b.
It is known that HCF of a and b = HCF of b and a for $a,b\in N$.
Therefore, $a\ast b=b\ast a$. Thus, the operation $\ast$ is commutative.
For $a,b,c\in N$, we have $(a\ast b)\ast c$ = (HCF of a and b) $\ast c=HCF$ of a,b and c and $a\ast (b\ast c)=a\ast$ (HCF of b and c) =HCF of a,b, and c
Therefore, $(a\ast b)\ast c=a\ast (b\ast c)$
Thus, the operation $\ast$ is associative.
Now, an element e in N will be the identity for the operation $\ast$ if $a\ast e=a=e\ast a$, for $\forall a\in N.$
But this relation is not true for any $a\in N.$
Thus, the operation $\ast$ does not have identity in N.