Let A be the set of natural numbers less than 6 and whose greatest common divisor with 6 is 1. Let B be the set of divisors of 6. What are the cardinalities of A, B, AUB and A∩B?

We are given two sets \( A \) and \( B \), and we are asked to determine the cardinalities of \( A \), \( B \), \( A \cup B \), and \( A \cap B \).

### Step 1: Define set \( A \)
Set \( A \) consists of natural numbers less than 6 whose greatest common divisor (GCD) with 6 is 1. The natural numbers less than 6 are:
\[
1, 2, 3, 4, 5
\]
Now, we check the GCD of each of these numbers with 6:
- \( \gcd(1, 6) = 1 \)
- \( \gcd(2, 6) = 2 \)
- \( \gcd(3, 6) = 3 \)
- \( \gcd(4, 6) = 2 \)
- \( \gcd(5, 6) = 1 \)

So, the numbers whose GCD with 6 is 1 are \( 1 \) and \( 5 \). Therefore, we have:
\[
A = \{1, 5\}
\]
The cardinality of \( A \) is:
\[
|A| = 2
\]

### Step 2: Define set \( B \)
Set \( B \) consists of divisors of 6. The divisors of 6 are:
\[
1, 2, 3, 6
\]
Therefore, we have:
\[
B = \{1, 2, 3, 6\}
\]
The cardinality of \( B \) is:
\[
|B| = 4
\]

### Step 3: Find \( A \cup B \) (Union of \( A \) and \( B \))
The union \( A \cup B \) consists of all elements that are in either \( A \) or \( B \). Combining the elements of \( A \) and \( B \):
\[
A \cup B = \{1, 5\} \cup \{1, 2, 3, 6\} = \{1, 2, 3, 5, 6\}
\]
The cardinality of \( A \cup B \) is:
\[
|A \cup B| = 5
\]

### Step 4: Find \( A \cap B \) (Intersection of \( A \) and \( B \))
The intersection \( A \cap B \) consists of elements that are in both \( A \) and \( B \). From the sets \( A = \{1, 5\} \) and \( B = \{1, 2, 3, 6\} \), the common element is \( 1 \):
\[
A \cap B = \{1\}
\]
The cardinality of \( A \cap B \) is:
\[
|A \cap B| = 1
\]

### Final Answers:
- \( |A| = 2 \)
- \( |B| = 4 \)
- \( |A \cup B| = 5 \)
- \( |A \cap B| = 1 \)

Would you like further details on any of the steps?

Here are 5 related questions to explore:
1. What is the greatest common divisor (GCD) of any two numbers?
2. How do you find the divisors of a number systematically?
3. Can you generalize the concept of GCD for different numbers like 8 or 12?
4. How do you calculate the union and intersection of sets involving more elements?
5. What are some properties of sets when considering unions and intersections?

**Tip:** When finding divisors of a number, always check the prime factorization—it helps in systematically listing divisors.

Learn how to find the cardinalities of sets A and B where A includes numbers whose GCD with 6 is 1, and B is the set of divisors of 6. Explore union and intersection operations in set theory, along with GCD concepts. Suitable for Grades 6-8.

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