In a two digit number, the unit's digit exceeds its ten's digit by 2. The product of the given number and the sum of its digits is equal to 144. Find the number.

Let ten's digit be x.

Unit's digit = x + 2

Number = 10x + x + 2 = 11x + 2

Sum of digits = x + x + 2 = 2x + 2

According to question,

Number × Sum of digits = 144

⇒ (11x + 2)(2x + 2) = 144

⇒ 22x² + 22x + 4x + 4 = 144

⇒ 22x² + 26x + 4 = 144

⇒ 22x² + 26x = 140

⇒ 2(11x² + 13x) = 140

⇒ 11x² + 13x = 70

⇒ 11x² + 13x - 70 = 0

⇒ 11x² + 35x - 22x - 70 = 0

⇒ x(11x + 35) - 2(11x + 35) = 0

⇒ (x - 2)(11x + 35) = 0

⇒ x - 2 = 0 or 11x + 35 = 0

⇒ x = 2 or x = -$\dfrac{35}{11}$

Since, digit cannot be negative.

∴ x = 2.

Number = 11x + 2 = 11(2) + 2

= 22 + 2 = 24.

Hence, number = 24.