Define * on N by m * n = 1 cm (m, n).

Solution:

$*$ is an operation as $m * n=\operatorname{LCM}(m, n)$ where $m, n \in N$. Let $m=2$ and $b=3$ two natural numbers.

m*n = 2*3

$=\operatorname{LCM}(2,3)$

$=6 \in \mathrm{N}$

So, $*$ is a binary operation from $\mathrm{N} \times \mathrm{N} \rightarrow \mathrm{N}$.

For commutative,

$\mathrm{n} * \mathrm{~m}=3 * 2$

$=\operatorname{LCM}(3,2)$

$=6 \in \mathrm{N}$

Since $m^{*} n=n^{*} m$, hence $*$ is commutative operation.

Again, for associative, let p = 4

$\mathrm{m} *(\mathrm{n} * \mathrm{p})=2 * \operatorname{LCM}(3,4)$

$=2 * 12$

$=\operatorname{LCM}(2,12)$

$=12 \in \mathrm{N}$

$(m * n) * p=\operatorname{LCM}(2,3) * 4$

$=6 * 4$

$=\operatorname{LCM}(6,4)$

$=12 \in \mathrm{N}$

As $m *(n * p)=(m * n) * p$, hence $*$ an associative operation.