If `A={x:x=3n,n in N} and B={x:x=4n,n in N}`, then find `A cap B`.

Answer

To find the intersection of the sets \( A \) and \( B \), we will first define the sets based on the given conditions and then determine their intersection.

### Step 1: Define the sets \( A \) and \( B \)

The set \( A \) is defined as:
\[ A = \{ x : x = 3n, n \in \mathbb{N} \} \]
This means \( A \) consists of all multiples of 3. For example, if we substitute values for \( n \):
- If \( n = 1 \), \( x = 3 \times 1 = 3 \)
- If \( n = 2 \), \( x = 3 \times 2 = 6 \)
- If \( n = 3 \), \( x = 3 \times 3 = 9 \)
- If \( n = 4 \), \( x = 3 \times 4 = 12 \)
- Continuing this way, we get \( A = \{ 3, 6, 9, 12, 15, 18, \ldots \} \)

The set \( B \) is defined as:
\[ B = \{ x : x = 4n, n \in \mathbb{N} \} \]
This means \( B \) consists of all multiples of 4. For example, if we substitute values for \( n \):
- If \( n = 1 \), \( x = 4 \times 1 = 4 \)
- If \( n = 2 \), \( x = 4 \times 2 = 8 \)
- If \( n = 3 \), \( x = 4 \times 3 = 12 \)
- If \( n = 4 \), \( x = 4 \times 4 = 16 \)
- Continuing this way, we get \( B = \{ 4, 8, 12, 16, 20, 24, \ldots \} \)

### Step 2: Find the intersection \( A \cap B \)

The intersection \( A \cap B \) consists of elements that are common to both sets \( A \) and \( B \).

To find the common elements, we look for numbers that are multiples of both 3 and 4. The least common multiple (LCM) of 3 and 4 is 12. Therefore, the common multiples will be:
- \( 12 \) (which is \( 3 \times 4 \))
- \( 24 \) (which is \( 12 \times 2 \))
- \( 36 \) (which is \( 12 \times 3 \))
- \( 48 \) (which is \( 12 \times 4 \))
- And so on...

Thus, the intersection can be expressed as:
\[ A \cap B = \{ x : x = 12n, n \in \mathbb{N} \} \]

### Final Answer

Therefore, the intersection \( A \cap B \) is:
\[ A \cap B = \{ 12, 24, 36, 48, \ldots \} = \{ 12n : n \in \mathbb{N} \} \]