Select a set A = {(x, y) ∈ N| НОД (x, y) = 7}, where N = {1, 2, 3, …}.

To select the set \( A = \{(x, y) \in \mathbb{N} \mid \gcd(x, y) = 7\} \), where \( \mathbb{N} = \{1, 2, 3, \dots\} \), we are looking for pairs of natural numbers \( (x, y) \) such that their greatest common divisor (GCD) is 7.

### Key observation:
- If \( \gcd(x, y) = 7 \), this means that both \( x \) and \( y \) must be multiples of 7. In other words, both \( x = 7a \) and \( y = 7b \), where \( a \) and \( b \) are natural numbers, and \( \gcd(a, b) = 1 \).
- The GCD of \( x \) and \( y \) is determined by their factors, and since the GCD is 7, this indicates that \( x \) and \( y \) share exactly 7 as a common divisor and no higher common factor.

### Expressing the set:
- Let \( x = 7a \) and \( y = 7b \), where \( \gcd(a, b) = 1 \) and \( a, b \in \mathbb{N} \).
- Thus, the set \( A \) can be written as:
  \[
  A = \{(7a, 7b) \mid a, b \in \mathbb{N}, \gcd(a, b) = 1\}.
  \]

In this form, \( A \) consists of all pairs of numbers that are multiples of 7, with the additional condition that the corresponding coefficients \( a \) and \( b \) must be coprime (i.e., \( \gcd(a, b) = 1 \)).

### Example pairs in \( A \):
- \( (7, 7) \) (since \( \gcd(1, 1) = 1 \)),
- \( (7, 14) \) (since \( \gcd(1, 2) = 1 \)),
- \( (21, 7) \) (since \( \gcd(3, 1) = 1 \)),
- \( (35, 14) \) (since \( \gcd(5, 2) = 1 \)), and so on.

Would you like more details on how to generate such pairs, or do you have any specific questions?

### Relative Questions:
1. What are the properties of the GCD that influence set \( A \)?
2. How do we prove that both \( x \) and \( y \) must be divisible by 7 in the given set?
3. Can you determine the structure of similar sets for \( \gcd(x, y) = k \) where \( k \neq 7 \)?
4. How do we generate all coprime pairs \( (a, b) \) efficiently?
5. Can the set \( A \) be represented graphically, and if so, how would it look?

### Tip:
When finding pairs of numbers with a specific GCD, factor both numbers and look at the highest common factor shared by both.

Select a set A = {(x, y) ∈ N | GCD(x, y) = 7}, where N = {1, 2, 3, …}.

Discover how to construct the set A = {(x, y) ∈ N | GCD(x, y) = 7}, exploring key number theory concepts such as GCD and coprime numbers. Learn how the GCD conditions lead to pairs of multiples of 7, where GCD(a, b) = 1. This problem is ideal for students at the undergraduate level or advanced high school level.

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