In a certain two digit number, the digit in the units place exceeds the digit in the tens place by 2. The product of the number and the sum of its digits is 144. What is the number?

Introduction / Context:
This problem is a digit based number puzzle. It describes a two digit number with a specific relationship between its tens and units digits, and a product condition involving the number and the sum of its digits. Our goal is to translate the conditions into algebraic equations and determine the original number. Such problems are common in quantitative aptitude sections and help build skill in working with place value and simple equations.

Given Data / Assumptions:
- The number is a two digit number with tens digit x and units digit y.
- The units digit y exceeds the tens digit x by 2, so y = x + 2.
- The number is therefore 10x + y.
- The sum of its digits is x + y.
- The product of the number and the sum of its digits equals 144.
- We must find the original two digit number.

Concept / Approach:
Using the digit relationship, we express y in terms of x. Then the number and the sum of its digits are both expressed as algebraic expressions in x. The given product condition yields an equation involving x only. Solving this equation gives the tens digit, and then the units digit follows from the relation y = x + 2. The actual two digit number is then formed as 10x + y.

Step-by-Step Solution:
Step 1: Let the tens digit be x and the units digit be y, so the number is 10x + y.Step 2: The units digit exceeds the tens digit by 2, so y = x + 2.Step 3: The sum of the digits is x + y. Using y = x + 2, this sum becomes x + (x + 2) = 2x + 2.Step 4: The product of the number and the sum of its digits is given as 144, so (10x + y) * (x + y) = 144.Step 5: Substitute y = x + 2 into the number: 10x + y becomes 10x + (x + 2) = 11x + 2.Step 6: The sum of the digits is 2x + 2, so the product condition becomes (11x + 2) * (2x + 2) = 144.Step 7: Expand the left hand side: (11x + 2)(2x + 2) = 22x^2 + 22x + 4x + 4 = 22x^2 + 26x + 4.Step 8: Set the expression equal to 144: 22x^2 + 26x + 4 = 144.Step 9: Subtract 144 from both sides to get 22x^2 + 26x - 140 = 0.Step 10: Divide the entire equation by 2 to simplify: 11x^2 + 13x - 70 = 0.Step 11: Solve this quadratic equation. One integer solution is x = 2.Step 12: With x = 2, y = x + 2 = 4, so the number is 10 * 2 + 4 = 24.

Verification / Alternative check:
Take the number 24 and check both conditions. The tens digit is 2 and the units digit is 4. The units digit exceeds the tens digit by 2, because 4 = 2 + 2. The sum of the digits is 2 + 4 = 6. The product of the number and the sum of its digits is 24 * 6 = 144, exactly matching the given condition. Therefore, 24 satisfies all requirements of the problem.

Why Other Options Are Wrong:
Consider 12, 36, 48, or 18. For 12, the units digit 2 does not exceed the tens digit 1 by 2. For 36, the units digit 6 exceeds the tens digit 3 by 3, not 2. For 48 and 18, the sum of digits multiplied by the number does not equal 144. For example, 36 * (3 + 6) = 36 * 9 = 324, which is not 144. None of these numbers satisfy both the digit difference and the product condition.

Common Pitfalls:
Common mistakes include misreading the phrase "exceeds by 2" and writing y = x - 2 instead of y = x + 2, or incorrectly forming the product equation. Some learners also try random numbers instead of forming and solving the equation, which can be time consuming and unreliable. Using a systematic algebraic approach is the most efficient and accurate method.

Final Answer:
The required two digit number is 24.