The product of the digits of a two digit number is 24. When 45 is added to the number, the digits interchange their places. What is the original number?

Introduction / Context:
This number puzzle combines the product of digits with a transformation that reverses the digits of a two digit number when a fixed amount is added. It is a common pattern in aptitude exams to test comfort with representing numbers using digits and setting up equations.

Given Data / Assumptions:
- The number is a two digit positive integer.
- The product of its tens and units digits is 24.
- Adding 45 to the original number produces another number with digits reversed.
- The goal is to determine the original two digit number.

Concept / Approach:
Represent the two digit number as 10x + y where x is the tens digit and y is the units digit. The reversed number then becomes 10y + x. Two conditions are given: xy = 24 for the product of digits and 10x + y + 45 = 10y + x for the reversal after addition. Solving this system of equations leads to the correct digits.

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 product condition gives xy = 24.Step 3: After adding 45, the number becomes 10x + y + 45, which equals the reversed number 10y + x.Step 4: Set up the equation: 10x + y + 45 = 10y + x.Step 5: Simplify: 10x - x + y - 10y + 45 = 0 which becomes 9x - 9y + 45 = 0, or 9(x - y + 5) = 0.Step 6: Thus x - y + 5 = 0, so x - y = -5 and x = y - 5.Step 7: Substitute into xy = 24 giving (y - 5) * y = 24.Step 8: Expand: y^2 - 5y - 24 = 0. Factorize to get (y - 8)(y + 3) = 0, so y = 8 or y = -3. Only y = 8 is valid for a digit.Step 9: Then x = y - 5 = 8 - 5 = 3, so the number is 10x + y = 10 * 3 + 8 = 38.

Verification / Alternative check:
Check the conditions. The digits of 38 are 3 and 8 with product 24, so the first condition is satisfied. Adding 45 gives 38 + 45 = 83, which is exactly the reversal of 38. This confirms that 38 is the correct number.

Why Other Options Are Wrong:
Option 54 has digits 5 and 4 with product 20, not 24. Option 83 has the right digits but in reversed order; it does not satisfy the condition that adding 45 reverses its own digits. Option 45 has digit product 20 and adding 45 gives 90, which is not a reversal. Option 68 has digit product 48, which also does not match the given product of 24.

Common Pitfalls:
Some learners miswrite the equation and use 10y + x + 45 as the result instead of 10x + y + 45, or they treat xy as x + y. Others forget that digits must be between 0 and 9 and might keep both algebraic roots of y. Being careful with the representation of digits and checking feasibility of solutions is essential.

Final Answer:
The original two digit number described in the question is 38.