Question

If * be a binary operation on $N$ defined by $a^{\ast }b=H.C.F.$ of $a$ & $b$, then check if * is commutative. Is * associative.

Answer 1

Yes the binary operation $\ast$ is commutative.

As H.C.F of $a,b=$ H.C.F. of $b,a$ for $a,b$ being natural numbers.

Also $\ast$ is associative as $a\ast (b\ast c)=(a\ast b)\ast c$ for $a,b,c$ being natural numbers.