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 :

Solution(By Examveda Team)

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
$$\eqalign{ & \therefore \left( {11x + 2} \right)\left( {2x + 2} \right) = 144 \cr & \Leftrightarrow 22{x^2} + 26x - 140 = 0 \cr & \Leftrightarrow 11{x^2} + 13x - 70 = 0 \cr & \Leftrightarrow \left( {x - 2} \right)\left( {11x + 35} \right) = 0 \cr & \Leftrightarrow x = 2 \cr} $$
Hence, required number:
= 11x + 2
= 11× 2 + 2
= 24