In a two digit number, the unit's digit exceeds ten's digit by $2$ and the product of the sum of the digits and the number itself is $144$, then what is the number?

Let the ten's digit be $t$ and the unit's digit be $u$.
Given that $u=t+2$.
The number is $10t+u=10t+(t+2)=11t+2$.
The sum of the digits is $t+u=t+(t+2)=2t+2$.

According to the question, the product of the sum of digits and the number is $144$:

$$(2t+2)(11t+2)=144$$

Let's check the options to find a match:

* Option A: $24$
Ten's digit ($t$) = $2$, Unit's digit ($u$) = $4$ (exceeds by 2 ✓)
Sum of digits = $2+4=6$
Product = $6\times 24=144$ (Matches exactly ✓)

Therefore, the number is $24$.