In a two-digit, if it is known that its unit's digit 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:

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.

 (11x + 2)(2x + 2) = 144

 22x² + 26x - 140 = 0

 11x² + 13x - 70 = 0

 (x - 2)(11x + 35) = 0

 x = 2.

Hence, required number = 11x + 2 = 24.