is anb a subset of c if a = {n e N: 2|n} and b =  {n e N: 3|n} and c=  {n e N: 6|n}

Let's define the sets step by step:

- **Set A**: \( A = \{ n \in \mathbb{N} : 2 \mid n \} \) — this means that \( A \) consists of all natural numbers divisible by 2, i.e., \( A = \{2, 4, 6, 8, 10, \dots \} \).

- **Set B**: \( B = \{ n \in \mathbb{N} : 3 \mid n \} \) — this means that \( B \) consists of all natural numbers divisible by 3, i.e., \( B = \{3, 6, 9, 12, 15, \dots \} \).

- **Set C**: \( C = \{ n \in \mathbb{N} : 6 \mid n \} \) — this means that \( C \) consists of all natural numbers divisible by 6, i.e., \( C = \{6, 12, 18, 24, \dots \} \).

Now, we need to find \( A \cap B \), which is the intersection of \( A \) and \( B \). This intersection will include all elements that are divisible by both 2 and 3.

Since the least common multiple (LCM) of 2 and 3 is 6, \( A \cap B \) will consist of all natural numbers divisible by 6:
\[
A \cap B = \{6, 12, 18, 24, \dots \}
\]

Finally, we check whether \( A \cap B \) is a subset of \( C \). Since \( C = \{6, 12, 18, 24, \dots \} \) consists of all multiples of 6, and \( A \cap B = \{6, 12, 18, 24, \dots \} \) is exactly the same set, it follows that:
\[
A \cap B \subseteq C
\]

Therefore, **yes**, \( A \cap B \) is a subset of \( C \).

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Do you want more details or have any questions?

Here are 5 related questions:
1. What is the union of sets \( A \) and \( B \) in this case?
2. How do we find the least common multiple (LCM) of two numbers?
3. Can the intersection of two sets ever be larger than one of the original sets?
4. What is the greatest common divisor (GCD) of 2 and 3, and how is it different from the LCM?
5. How would the answer change if \( C \) was \( \{n \in \mathbb{N} : 12 \mid n \} \)?

**Tip:** When finding intersections of sets defined by divisibility, look for the least common multiple of the divisors involved!

Is A ∩ B a subset of C if A = {n ∈ N : 2|n}, B = {n ∈ N : 3|n}, and C = {n ∈ N : 6|n}?

Explore the problem of determining whether the intersection of sets A and B, where A is divisible by 2 and B is divisible by 3, is a subset of C, a set divisible by 6. This solution covers key set theory concepts, including intersections, divisibility, and least common multiples. Suitable for students in Grades 9-12.

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