Decoding the Two-Digit Number Mystery

Mathematics is a treasure trove of riddles and puzzles. Today, we delve into one such intriguing question that involves a two-digit number and its unique properties. This article explores the solution to the riddle and provides a detailed explanation for each step of the journey to find the number.

The Riddle: A Product and a Reversal

The challenge lies in identifying a two-digit number where the product of its two digits equals 20, and when 9 is added to the number, the digits are reversed. This article will guide you through this mathematical puzzle.

Solving the Puzzle

First, let's break down the problem into simpler components. We are dealing with a two-digit number, where the digits are X (tens place) and Y (units place). The number can be represented as 1 Y and its reverse as 10Y X.

The problem statement provides us with two key pieces of information:

The product of the digits (X * Y) is 20. Adding 9 to the number results in its reverse: (1 Y) 9 10Y X.

Let's start with the second condition:

Step 1: Setting Up the Equation

By simplifying the second condition, we get:

$$1 Y 9 10Y X$$

Rearranging the terms, we have:

$$1 - X Y - 10Y -9$$ $$9X - 9Y -9$$

Dividing both sides by 9:

$$X - Y -1$$

Or, equivalently:

$$Y X 1$$

Step 2: Finding the Digits

Looking at the product condition, X * Y 20, we can substitute Y from the previous equation:

$$X * (X 1) 20$$

Expanding and rearranging gives us:

$$X^2 X - 20 0$$

Solving this quadratic equation, we use the quadratic formula:

$$X frac{-b pm sqrt{b^2 - 4ac}}{2a}$$

Here, a 1, b 1, and c -20. Plugging these values in:

$$X frac{-1 pm sqrt{1 80}}{2} frac{-1 pm sqrt{81}}{2} frac{-1 pm 9}{2}$$

This gives us two possible solutions for X:

$$X frac{8}{2} 4$$ $$X frac{-10}{2} -5$$

Since X represents a digit in a two-digit number, it must be a positive integer. Therefore, X 4. Substituting X into the equation Y X 1, we find:

$$Y 4 1 5$$

The digits are X 4 and Y 5.

Thus, the original number is 45, and its reverse is 54. Adding 9 to 45 gives us 54, which satisfies the original condition.

Conclusion

The puzzle reveals that the two-digit number, where the product of its digits is 20, and adding 9 results in its reverse, is 45. This solution was derived by setting up equations based on the given conditions and solving them step-by-step.

Understanding such puzzles enhances critical thinking and problem-solving skills. Math is not just about numbers; it is a fascinating domain that requires creativity and logical reasoning.

References:

[Reference 1]

[Reference 2]