Final answer:
The question involves solving a mathematical problem where a two-digit number must satisfy two conditions: the unit's digit is two more than the ten's digit, and the product of the number and the sum of its digits is 144. Upon setting up the equation (10x + x + 2)(2x + 2) = 144 and solving it, we determine that the number is 24. Hence, the correct answer is a. 24.
Step-by-step explanation:
To find the two-digit number of which the unit’s digit exceeds the ten’s digit by 2 and the product of this number and the sum of its digits is 144, we can set up an algebraic equation. Let the ten’s digit be x, then the unit’s digit would be x + 2. Our two-digit number can be represented as 10x + (x + 2). The sum of the digits is x + (x + 2) which simplifies to 2x + 2. Given that the product of the number and the sum of its digits is 144, we have the following equation:
(10x + x + 2)(2x + 2) = 144.
Expanding this equation, we get:
(11x + 2)(2x + 2) = 144,
22x2 + 4x + 44 = 144,
22x2 + 4x - 100 = 0,
11x2 + 2x - 50 = 0.
By solving this quadratic equation, we find that x = 2 (since the number must be positive and a two-digit number). Therefore, the unit’s digit is 2 + 2 = 4 and the ten’s digit is 2. So the number is 24.
The correct answer from the options given is a. 24.