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.