Solving a Mathematical Brain Teaser: Finding a Two-Digit Number with Specific Properties

One of the most delightful aspects of mathematics is solving brain teasers and number puzzles. Today, we will explore a classic example of such a teaser. The problem involves a two-digit number whose digits have a specific relationship: the product of the digits is 20, and if you add 9 to the number, the digits interchange their places. Let's dive into the solution and the underlying principles that make this puzzle so interesting.

The Brain Teaser

Consider the following teaser:

A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. What is the number?

Solution Approach

To solve this problem, let's denote the two-digit number as 10t u, where t is the tens digit and u is the units digit. Let's break down the problem into two parts: The product of the digits is 20. If 9 is added to the number, the digits interchange their places.

Step 1: The Product of the Digits

The first condition states that the product of the digits is 20. Therefore, we can write the equation:

tu 20

Step 2: Interchanging the Digits

The second condition tells us that if we add 9 to the number, the digits interchange their places. This can be expressed as:

10t u 9 10u t

Let's simplify this equation:

10t u 9 - 10u - t 0
9t - 9u 9 0
t - u 1 0
t u - 1

Step 3: Solving the Equations

Now we can substitute the value of t from the second equation into the first equation:

(u - 1)u 20
u^2 - u - 20 0

We can solve this quadratic equation using the quadratic formula or by factoring:

(u - 5)(u 4) 0

This gives us two possible solutions for u: u 5 or u -4. Since u must be a digit (0 to 9), we discard u -4 and get u 5.

Substitute u 5 into the equation t u - 1:

t 5 - 1 4

Thus, the number is 45, which satisfies both conditions of the problem.

Understanding the Solution

Let's verify the solution to ensure it meets all conditions:

The number 45 is the correct answer.

General Insight

This brain teaser showcases how mathematical principles can be used to solve complex problems. Understanding the relationships between digits and their positions is key in solving such teasers. Moreover, the problem requires the application of algebraic concepts such as equations and quadratic equations.

Conclusion

The problem of finding a two-digit number whose digits have a specific relationship is a fascinating brain teaser. It not only tests one's mathematical skills but also enhances problem-solving abilities. By breaking down the problem into manageable parts, we can use algebraic methods to derive the solution. Such puzzles are not only fun but also enrich our understanding of mathematics.