In a two-digit number, if it is known that its unit's digits exceeds its ten's digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is :

Answer: Option A

Let the ten's digit be x
Then, unit's digit = x + 2
∴ Number :
= 10x + (x + 2)
= 11x + 2
Sum of digits :
= x + (x + 2)
= 2x + 2
$$\begin{array}{rl}& \therefore \left(11x+2\right)\left(2x+2\right)=144\\ & \iff 22x^2+26x-140=0\\ & \iff 11x^2+13x-70=0\\ & \iff \left(x-2\right)\left(11x+35\right)=0\\ & \iff x=2\end{array}$$
Hence, required number:
= 11x + 2
= 11× 2 + 2
= 24