Let $*$ be the binary operation on N given by a$*$ b$=$ L.C.M. of a and b. Find

(i) 5 $*$ 7, 20 $*$ 16

(ii) Is $*$ commutative ?

(iii) Is $*$ associative ?

(iv) Find the identity of $*$ in N

(v) Which elements of N are invertible for the operation $*$ ?

a $*$ b $=$ L.C.M. of a and b.

(i) 5 $*$ 7 $=$ L.C.M. of 5 and 7 $=$ 35

20 $*$ 16 $=$ L.C.M. of 20 and 16 $=$ 80

(ii) a $*$ b $=$ L.C.M. of a and b $=$ L.C.M. of b and a $=$ b $*$ a.

Thus, operation $*$ is commutative.

(iii) a $*$ (b $*$ c) $=$ a $*$ (L.C.M. of b and c)

$=$ L.C.M. of (a and (L.C.M. of b and c))

$=$ L.C.M. of a, b and c.

Similarly, (a $*$ b) $*$ c $=$ (L.C.M. of a and b) $*$ c

$=$ L.C.M. of ((L.C.M. of a and b) and c) $=$ L.C.M. of a, b and c

Thus, a $*$ (b $*$ c) $=$ (a $*$ b ) $*$ c

Hence, the operation $*$ is associative.

(iv) Identity of $*$ in N $=$ 1 because, a $*$ 1

$=$ L.C.M. of a and 1 $=$ 1

(v) Only the element 1 in N is invertible for the operation $*$ because $1 * \cfrac{1}{1} = 1$.