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 $$N$$?

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

Toolbox:
• 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 $a \in N$
• The highest common factor (HCF), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.
Given that the binary operation on N given by $a \ast b =$ HCF $(a,b)$:
$\textbf {Step 1: Checking if the operation is commutative}$: