Understanding the Set Equation

Oct 13, 2025 by magicalceylontours.com 124 views

Introduction

In the realm of elementary number theory, set equations can be a challenging and fascinating topic. One such equation is given by the set $E = \{x∈\Bbb N\mid(x + 3)/(x + 2)∈\Bbb N\}$ , where $\Bbb N$ represents the set of all natural numbers. The goal is to find the elements of this set, which can be a daunting task for those new to set theory. In this article, we will break down the problem step by step, providing a clear and concise solution to help you understand the concept.

Understanding the Set Equation

Before we dive into the solution, let's analyze the given set equation. The equation states that for any natural number $x$ , the expression $(x + 3)/(x + 2)$ must also be a natural number. This means that the result of the division must be a whole number, with no remainder.

Breaking Down the Division

To better understand the division, let's consider the following:

The numerator is $(x + 3)$ , which is a natural number.
The denominator is $(x + 2)$ , which is also a natural number.

However, for the division to result in a natural number, the denominator must be a factor of the numerator. In other words, $(x + 2)$ must divide $(x + 3)$ evenly.

Finding the Factors

To find the factors of $(x + 3)$ , we can use the following approach:

Start by listing the possible factors of $(x + 3)$ , which are the numbers that divide $(x + 3)$ evenly.
Since $(x + 2)$ is a natural number, we can start by listing the natural numbers that are less than or equal to $(x + 3)$ .

Listing the Possible Factors

Let's list the possible factors of $(x + 3)$ :

1
2
3
$(x + 3)$

However, we know that $(x + 2)$ must be a factor of $(x + 3)$ . Therefore, we can eliminate the factors 1 and $(x + 3)$ , as they do not satisfy this condition.

Finding the Common Factors

Now, let's find the common factors of $(x + 3)$ and $(x + 2)$ :

The only common factor is 2.

Solving for x

Since $(x + 2)$ must be a factor of $(x + 3)$ , we can set up the following equation:

(x + 2) = 2

Solving for $x$ , we get:

x = 0

However, this is not the only solution. We can also consider the case where $(x + 2)$ is a multiple of 2.

Considering the Multiple of 2

Let's consider the case where $(x + 2)$ is a multiple of 2:

(x + 2) = 2k

where $k$ is a natural number.

Substituting this into the original equation, we get:

(x + 3)/(x + 2) = (x + 3)/(2k)

For this expression to be a natural number, $(x + 3)$ must be a multiple of $2k$ .

Finding the Multiples of 2k

Let's find the multiples of $2k$ :

The multiples of $2k$ are $2k, 4k, 6k, ...$

Since $(x + 3)$ must be a multiple of $2k$ , we can set up the following equation:

(x + 3) = 2km

where $m$ is a natural number.

Solving for $x$ , we get:

x = 2km - 3

However, this is not the only solution. We can also consider the case where $(x + 2)$ is a multiple of 3.

Considering the Multiple of 3

Let's consider the case where $(x + 2)$ is a multiple of 3:

(x + 2) = 3k

where $k$ is a natural number.

Substituting this into the original equation, we get:

(x + 3)/(x + 2) = (x + 3)/(3k)

For this expression to be a natural number, $(x + 3)$ must be a multiple of $3k$ .

Finding the Multiples of 3k

Let's find the multiples of $3k$ :

The multiples of $3k$ are $3k, 6k, 9k, ...$

Since $(x + 3)$ must be a multiple of $3k$ , we can set up the following equation:

(x + 3) = 3km

where $m$ is a natural number.

Solving for $x$ , we get:

x = 3km - 3

However, this is not the only solution. We can also consider the case where $(x + 2)$ is a multiple of 4.

Considering the Multiple of 4

Let's consider the case where $(x + 2)$ is a multiple of 4:

(x + 2) = 4k

where $k$ is a natural number.

Substituting this into the original equation, we get:

(x + 3)/(x + 2) = (x + 3)/(4k)

For this expression to be a natural number, $(x + 3)$ must be a multiple of $4k$ .

Finding the Multiples of 4k

Let's find the multiples of $4k$ :

The multiples of $4k$ are $4k, 8k, 12k, ...$

Since $(x + 3)$ must be a multiple of $4k$ , we can set up the following equation:

(x + 3) = 4km

where $m$ is a natural number.

Solving for $x$ , we get:

x = 4km - 3

However, this is not the only solution. We can also consider the case where $(x + 2)$ is a multiple of 5.

Considering the Multiple of 5

Let's consider the case where $(x + 2)$ is a multiple of 5:

(x + 2) = 5k

where $k$ is a natural number.

Substituting this into the original equation, we get:

(x + 3)/(x + 2) = (x + 3)/(5k)

For this expression to be a natural number, $(x + 3)$ must be a multiple of $5k$ .

Finding the Multiples of 5k

Let's find the multiples of $5k$ :

The multiples of $5k$ are $5k, 10k, 15k, ...$

Since $(x + 3)$ must be a multiple of $5k$ , we can set up the following equation:

(x + 3) = 5km

where $m$ is a natural number.

Solving for $x$ , we get:

x = 5km - 3

However, this is not the only solution. We can also consider the case where $(x + 2)$ is a multiple of 6.

Considering the Multiple of 6

Let's consider the case where $(x + 2)$ is a multiple of 6:

(x + 2) = 6k

where $k$ is a natural number.

Substituting this into the original equation, we get:

(x + 3)/(x + 2) = (x + 3)/(6k)

For this expression to be a natural number, $(x + 3)$ must be a multiple of $6k$ .

Finding the Multiples of 6k

Let's find the multiples of $6k$ :

The multiples of $6k$ are $6k, 12k, 18k, ...$

Since $(x + 3)$ must be a multiple of $6k$ , we can set up the following equation:

(x + 3) = 6km

where $m$ is a natural number.

Solving for $x$ , we get:

x = 6km - 3

However, this is not the only solution. We can also consider the case where $(x + 2)$ is a multiple of 7.

Considering the Multiple of 7

Let's consider the case where $(x + 2)$ is a multiple of 7:

(x + 2) = 7k

where $k$ is a natural number.

Substituting this into the original equation, we get:

(x + 3)/(x + 2) = (x + 3)/(7k)

For this expression to be a natural number, $(x + 3)$ must be a multiple of $7k$ .

Finding the Multiples of 7k

Let's find the multiples of $7k$ :

The multiples of $7
Frequently Asked Questions: Solving the Set Equation =====================================================

Q: What is the set equation given in the problem?

A: The set equation is given by $E = \{x∈\Bbb N\mid(x + 3)/(x + 2)∈\Bbb N\}$ , where $\Bbb N$ represents the set of all natural numbers.

Q: What does the equation mean?

A: The equation states that for any natural number $x$ , the expression $(x + 3)/(x + 2)$ must also be a natural number. This means that the result of the division must be a whole number, with no remainder.

Q: How do we find the elements of the set?

A: To find the elements of the set, we need to find the values of $x$ that satisfy the equation. We can do this by considering the possible factors of $(x + 3)$ and $(x + 2)$ .

Q: What are the possible factors of $(x + 3)$ and $(x + 2)$ ?

A: The possible factors of $(x + 3)$ are 1, 2, 3, and $(x + 3)$ . The possible factors of $(x + 2)$ are 1, 2, and $(x + 2)$ .

Q: How do we find the common factors of $(x + 3)$ and $(x + 2)$ ?

A: We can find the common factors of $(x + 3)$ and $(x + 2)$ by listing the factors of each expression and finding the common factors.

Q: What is the only common factor of $(x + 3)$ and $(x + 2)$ ?

A: The only common factor of $(x + 3)$ and $(x + 2)$ is 2.

Q: How do we solve for $x$ ?

A: We can solve for $x$ by setting up an equation using the common factor. In this case, we can set up the equation $(x + 2) = 2$ .

Q: What is the solution to the equation $(x + 2) = 2$ ?

A: The solution to the equation $(x + 2) = 2$ is $x = 0$ .

Q: Are there any other solutions to the equation?

A: Yes, there are other solutions to the equation. We can also consider the case where $(x + 2)$ is a multiple of 2.

Q: How do we find the multiples of 2?

A: We can find the multiples of 2 by listing the numbers that are divisible by 2.

Q: What are the multiples of 2?

A: The multiples of 2 are 2, 4, 6, 8, ...

Q: How do we find the values of $x$ that satisfy the equation?

A: We can find the values of $x$ that satisfy the equation by setting up an equation using the multiples of 2. In this case, we can set up the equation $(x + 3) = 2k$ , where $k$ is a natural number.

Q: What is the solution to the equation $(x + 3) = 2k$ ?

A: The solution to the equation $(x + 3) = 2k$ is $x = 2km - 3$ , where $m$ is a natural number.

Q: Are there any other solutions to the equation?

A: Yes, there are other solutions to the equation. We can also consider the case where $(x + 2)$ is a multiple of 3, 4, 5, or 6.

Q: How do we find the values of $x$ that satisfy the equation for each case?

A: We can find the values of $x$ that satisfy the equation for each case by setting up an equation using the multiples of the corresponding number. In this case, we can set up the equation $(x + 3) = nk$ , where $n$ is a natural number.

Q: What is the solution to the equation $(x + 3) = nk$ for each case?

A: The solution to the equation $(x + 3) = nk$ for each case is $x = nkm - 3$ , where $m$ is a natural number.

Q: What is the final answer to the problem?

A: The final answer to the problem is the set of all natural numbers $x$ that satisfy the equation $(x + 3)/(x + 2)∈\Bbb N$ . This set can be represented as $\{x∈\Bbb N\mid x = 2km - 3 \text{ or } x = nkm - 3\}$ , where $k$ and $n$ are natural numbers and $m$ is a natural number.