Solving Equations and Factorization in Number Theory

The problem of finding integer solutions to equations often employs a combination of algebraic manipulation and number theory techniques. In this article, we delve into the process of solving a specific equation through factorization and factor pair analysis. The focus is on understanding the equation-solving and factorization methods, which are fundamental in number theory. This article is designed to be accessible to learners and experts alike, providing a comprehensive guide to solving equations and understanding the intricacies of factorization.

Introduction to the Problem

The initial equation given is:

[ x cdot y - 9x - 5y 64 ]

To simplify this equation, we follow the steps:

1. Add (5 cdot 9 45) to both sides:

[ xy - 9x - 5y 45 64 45 ]

2. Factor the left side to get:

[ (x - 5)(y - 9) 64 ]

This step is crucial as it transforms the original equation into a form that is more straightforward to analyze.

Understanding the Factorization

The factorization of 64 is essential. Since (64 2^6), the possible values for ((x - 5)) and ((y - 9)) are the factors of 64. Specifically, these factors can be: [ pm 1, pm 2, pm 4, pm 8, pm 16, pm 32, pm 64 ]

Each pair of factors can be used to find corresponding values for (x) and (y).

Identifying Valid Solutions

Given that ((x - 5)(y - 9) 64), we need to pair these factors. For each pair, we determine the integer values for (x) and (y).

For example:

((x - 5, y - 9) (1, 64)) implies ((x, y) (6, 73)) ((x - 5, y - 9) (2, 32)) implies ((x, y) (7, 41))

There are 14 such pairs, resulting in 14 integer solutions in total. This is because each of the 7 positive factor pairs corresponds to a positive and a negative solution for each corresponding variable.

Conclusion

The problem of solving the equation ( x cdot y - 9x - 5y 64 ) involves a systematic approach to factorization and understanding the properties of number pairs. The key steps are as follows:

Add a constant to both sides to simplify the equation. Factor the resulting expression. Identify all factor pairs of the constant. Use these pairs to find all corresponding solutions.

This method can be generalized to solve similar equations, making it a valuable technique in number theory and algebra.