Solving the Equation xyz * x * y * z xyz: The Search for Integer Solutions

Introduction

The equation in question, xyz * x * y * z xyz, presents a challenge to solve for integer values of x, y, and z. This exploration will delve into the methods and reasoning to determine the possible integer solutions to this equation, particularly for non-negative and all integers.

Transformation and Non-Negative Integer Solutions

To start, we rearrange the equation to simplify our analysis:

x * y * z * x * y * z xyz can be simplified to:

(x * y * z)^2 xyz

Further simplifying, we get:

x * y * z 1

This formulation allows us to explore the integer solutions more straightforwardly.

Non-Negative Integer Solutions

When considering non-negative integers, the equation x * y * z 1 can be viewed as the problem of distributing 1 indistinguishable object into 3 distinguishable boxes. Each box represents x, y, and z.

The number of such distributions is given by the binomial coefficient:

#x03C3;((n k - 1) / (k - 1))

For this equation, n 1 (the single object to distribute) and k 3 (the boxes x, y, and z).

#x03C3;((1 3 - 1) / (3 - 1)) #x03C3;(3 / 2) 3

The non-negative integer solutions are:

(1, 0, 0) (0, 1, 0) (0, 0, 1)

All Integer Solutions

For all integers, we consider the general case where x, y, and z can be positive, negative, or zero. We rewrite the equation for z in terms of x and y:

z 1 - x - y

For z to be an integer, x and y must also be integers. Thus, we can iterate over integer values of x and compute corresponding values of y and z.

If x and y are both non-negative, we have already found the non-negative solutions. If x or y is negative, we consider the range of possible values. For example: Let x 0. Then y * z 1, which gives solutions (0, 1, 0) and (0, 0, 1). Let x 1. Then y * z 0, which gives solutions (1, 0, 0), (1, 1, 0) and (1, 0, 1). Let x -1. Then y * z 2, which gives solutions such as (-1, 0, 2), (-1, 2, 0), (-1, 1, 1).

By continuing this process, we observe that x can take on negative values, leading to corresponding values for y and z. This results in infinitely many integer combinations.

Conclusion

From this analysis, we conclude that the equation xyz * x * y * z xyz has:

3 non-negative integer solutions: (1, 0, 0), (0, 1, 0), (0, 0, 1). Infinitely many integer solutions overall, as x can be any integer, leading to corresponding values for y and z.

The final answer is:

Infinitely many integer solutions exist.