Relations and Functions — Class 12 Maths Solution

Commutativity

a$*$ b $=$ H.C.F. of a and b $=$ H.C.F. of b and a $=$ b $*$ a

Thus, operation $*$ is commutative.

Associativity

(a $*$ b ) $*$ c $=$ (H.C.F. of a and b ) $*$ c

$=$ H.C.F. of [(H.C.F. of a and b) and c] $=$ H.C.F. a, b and c

a $*$ (b $*$ c ) $=$ a $*$ [H.C.F. of b and c]

$=$ H.C.F. of [a and (H.C.F. of b and c)] $=$ H.C.F. of [a, b and c]

$\Rightarrow$ (a $*$ b) $*$ c $=$ a $*$
(b $*$ c). Thus, operation $*$ is associative.

Identity

Now, 1 $*$ a $=$ a $*$ 1 $\ne$ a

There does not exist any identity element.