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

$\begin{array}{1 1} \text{Not Commutative, Not Associative, Identity Exists} \\ \text{Not Commutative, Not Associative, Identity Does not Exist} \\ \text{Commutative, Associative, Identity Exists} \\ \text{Commutative, Associative, Identity Does not Exist} \end{array}$

• An operation $\ast$ on $A$ is commutative if $a\ast b = b \ast a\; \forall \; a, b \in A$
• An operation $\ast$ on $A$ is associative if $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
• An element $e \in N$ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
Given that the binary operation on N given by $a \ast b =$ HCF $(a,b)$:
$\textbf {Step 1: Checking if the operation is commutative}$:
We know $HCF$(a,b) = $HCF$(b,a)$.$\Rightarrow a*b=b*a$for all$a,b \in N \rightarrow \ast $is commutative.$\textbf {Step 2: Checking if the operation is associative}$: For an operation$\ast$to be associative$a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in N(a*b)*c = $HCF$(a,b) \ast c =$HCF$(a,b,c) a*(b*c)= a*($HCF$(b,c)) =$HCF$(a,b,c)\Rightarrow (a*b)*c=a*(b*c) \forall a,b,c \in N \rightarrow \ast $is associative.$\textbf {Step 2: Checking if the operation has an identity}$: We know that the element$e \in N $is an identify element for operation * if$a*e=e*a$for all$a \in N$We can clearly see that this is NOT true for all$a \in N$through a small example: Let$a=5$,$ 5 \ast 1 = 1$and so is$6 \ast 1 = 1$. So, this relation is not true for al$a \in N$. Therefore$\ast$does not have an identity element in$N\$.