Question

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 $N$?

Answer 1

Check commutative

$\ast$ is commutative if $a\ast b=b\ast a$

$a\ast b=HCFofa\&bb\ast a=HCFofb\&a$

Since $a\ast b=b\ast a\forall a,bϵN$

$\ast$ is commutative.

Check associative

$\ast$ is associative if $(a\ast b)\ast c=a\ast (b\ast c)$

$(a\ast b)\ast c=\left(HCFofa\&b\right)\ast c=HCFof\left(HCFofa\&b\right)\&c=HCFofa,b\&ca\ast (b\ast c)=a\ast \left(HCFofb\&c\right)=HCfofa\&\left(HCFofb\&c\right)=HCFofa,b\&c$

Since,

$(a\ast b)\ast c=a\ast (b\ast c)\forall a,bϵN$

$\ast$ is associative.

Identity Element

$e$ is the identity element of $\ast$ if

$a\ast e=e\ast a=a$

i.e.,$HCFofa$ and $e=HCFofe$ and $a=a$

there is no value of $e$ which satisfies the given condition.

$eg:lete=1$

$HCFofa$ and$1=1\ne a$

$HCFof1$ and$a=1\ne a$

thus,there is no identity of $\ast$ in $N$.