The product of the digits of a two digit number is 15. If 18 is added to the number, the new number formed has its digits interchanged compared to the original number. What is the original two digit number?

Introduction / Context:
This number puzzle focuses on two digit numbers and their digits. The problem tells you the product of the digits and also describes what happens when a constant is added to the number. These conditions together uniquely determine the original number. Problems of this kind test your ability to form and solve equations based on digit relationships.

Given Data / Assumptions:
- The number is a two digit number.
- Let the tens digit be x and the units digit be y.
- The product of the digits is x * y = 15.
- When 18 is added to the original number, the digits are interchanged to form the new number.
- All digits are from 0 to 9, but since it is a two digit number, the tens digit is from 1 to 9.
- We must find the original two digit number.

Concept / Approach:
Represent the two digit number as 10x + y. When the digits are reversed, the new number becomes 10y + x. The condition about adding 18 gives a linear equation connecting these two expressions. Combining this with the information that the product of the digits is 15 allows us to solve systematically for x and y. By checking integer factor pairs of 15, we avoid unnecessary algebraic complexity.

Step-by-Step Solution:
Step 1: Let the original number be 10x + y where x is the tens digit and y is the units digit.Step 2: The product of the digits is x * y = 15.Step 3: When 18 is added, the new number becomes 10x + y + 18.Step 4: This new number equals the number with digits interchanged, which is 10y + x.Step 5: Set up the equation: 10x + y + 18 = 10y + x.Step 6: Rearrange: 10x - x + y - 10y + 18 = 0, which simplifies to 9x - 9y + 18 = 0.Step 7: Divide the entire equation by 9 to get x - y + 2 = 0, or y - x = 2.Step 8: We now have two conditions: x * y = 15 and y - x = 2.Step 9: Possible positive factor pairs for 15 are (1, 15), (3, 5), (5, 3), and (15, 1).Step 10: We need y - x = 2. The pair x = 3, y = 5 satisfies this because 5 - 3 = 2.Step 11: Therefore, the original number is 10x + y = 10 * 3 + 5 = 35.

Verification / Alternative check:
Check the product of the digits: 3 * 5 = 15, which matches the given condition. Now add 18 to the original number: 35 + 18 = 53. The number 53 is indeed formed by interchanging the digits of 35. Both key conditions are satisfied, which confirms that 35 is the correct original number.

Why Other Options Are Wrong:
Numbers such as 15, 51, 21, and 53 either do not have a digit product of 15 or do not produce a digit reversed number when 18 is added. For example, 51 has digit product 5 * 1 = 5, not 15. Adding 18 to 21 gives 39, which is not the reverse of 21. Therefore, these options fail one or both of the required conditions.

Common Pitfalls:
Many learners try to guess numbers randomly instead of using the two algebraic conditions cleverly. Others may mistakenly set up the equation 10y + x + 18 = 10x + y, reversing the direction of the transformation. It is important to clearly define which number is original and which is formed after adding 18, and then consistently translate this into equations.

Final Answer:
The original two digit number is 35.