Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is the minimum product?

Key Concepts

Difference of Numbers

The difference of two numbers refers to the result obtained when one number is subtracted from another. In this problem, we are given that the difference between two numbers is 24, which can be expressed mathematically as x - y = 24, where x and y are the two numbers. Understanding this relationship is crucial for setting up the equations needed to find the numbers.

Product of Numbers

The product of two numbers is the result of multiplying them together. In this context, we are tasked with minimizing the product of the two numbers, which can be represented as P = x * y. To find the minimum product, we need to express one variable in terms of the other using the difference constraint and then analyze the resulting function.

Optimization

Optimization in mathematics involves finding the maximum or minimum value of a function. In this case, we are looking to minimize the product of two numbers under a specific constraint (their difference). Techniques such as substitution and calculus can be employed to determine the minimum product effectively, ensuring we understand how to analyze the function derived from the given conditions.