The product of the digits of a two-digit number is 27. If 54 is added to the number, the resulting number has its digits interchanged. What is the original two-digit number?

Introduction / Context:
This question combines number properties with a bit of algebra. You are told that a certain two digit number has digits whose product is 27. When 54 is added to this number, the digits of the new number are the reverse of the original digits. You must find the original two digit number. Problems like this test understanding of place value and how to represent a two digit number algebraically in terms of its tens and units digits.

Given Data / Assumptions:

- Let the two digit number be 10a + b, where a is the tens digit and b is the units digit. - a is a digit from 1 to 9, and b is a digit from 0 to 9. - The product of the digits is a * b = 27. - Adding 54 to the original number gives a new number whose digits are interchanged, that is 10b + a.

Concept / Approach:
We use two conditions: one for the product of the digits and another for the digit reversal after adding 54. The product condition restricts the possible pairs (a, b) to those whose product is 27. The reversal condition becomes a linear equation in a and b based on the decimal representation of the numbers. By solving this system of conditions, we identify the correct pair of digits and hence the original two digit number. This approach is systematic and avoids guesswork.

Step-by-Step Solution:
Step 1: From the product condition a * b = 27, list all possible integer digit pairs (a, b) with a from 1 to 9 and b from 0 to 9. Step 2: Factor 27. Possible pairs are (3, 9) and (9, 3). The combinations (1, 27) or (27, 1) are not valid digit pairs. Step 3: Now apply the reversal condition. The original number is 10a + b. After adding 54, the new number is 10b + a. Step 4: Write the equation: 10a + b + 54 = 10b + a. Step 5: Simplify: 10a + b + 54 = 10b + a implies 9a - 9b = -54. Step 6: Divide both sides by 9: a - b = -6, so a = b - 6. Step 7: Consider the first candidate pair from the product condition: (a, b) = (3, 9). Here a - b = 3 - 9 = -6, which matches the equation. Step 8: Consider the second candidate: (a, b) = (9, 3). Here a - b = 9 - 3 = 6, which does not match -6, so this pair is invalid. Step 9: Therefore, the digits must be a = 3 and b = 9, and the original number is 10a + b = 10 * 3 + 9 = 39.

Verification / Alternative check:
Check both conditions for the number 39. The digits are 3 and 9, whose product is 3 * 9 = 27, satisfying the first condition. Next, add 54: 39 + 54 = 93. The new number 93 has digits 9 and 3, which are indeed the reverse of the original digits. This confirms that 39 is the correct number and that no other candidate satisfies both conditions simultaneously.

Why Other Options Are Wrong:
Option B (93) is actually the result after adding 54, not the original number. Options C (63) and D (36) have digit products 18 and 18 respectively, not 27, so they fail the first condition. Even if one tried to apply the reversal condition, adding 54 to these numbers does not produce a number with digits reversed. Only 39 satisfies all the given requirements.

Common Pitfalls:
Some learners may forget that the tens digit cannot be zero or may incorrectly interpret "product of digits" as something involving the whole number. Others may assume any pair of digits with product 27 will work without testing the addition condition. Always turn the wording into clear algebraic equations and check each candidate pair carefully. This ensures a reliable solution without confusion.

Final Answer:
The original two digit number is 39, which corresponds to option A.