If `A={x:x=3n,n epsilon Z}` and `B={x:x=4n,n epsilon Z}`, then find `A nn B`.

Answer

To find the intersection of the sets \( A \) and \( B \), we will follow these steps:

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

The set \( A \) is defined as:
\[ A = \{ x : x = 3n, n \in \mathbb{Z} \} \]
This means that \( A \) consists of all multiples of 3, which includes both positive and negative integers, as well as zero.

The set \( B \) is defined as:
\[ B = \{ x : x = 4n, n \in \mathbb{Z} \} \]
This means that \( B \) consists of all multiples of 4, which also includes both positive and negative integers, as well as zero.

### Step 2: Identify the common elements in \( A \) and \( B \)

To find the intersection \( A \cap B \), we need to determine the elements that are common to both sets. This means we are looking for numbers that are multiples of both 3 and 4.

### Step 3: Find the least common multiple (LCM)

The common multiples of 3 and 4 can be found using the least common multiple (LCM).

To calculate the LCM of 3 and 4:
- The prime factorization of 3 is \( 3^1 \).
- The prime factorization of 4 is \( 2^2 \).

The LCM is found by taking the highest power of each prime:
\[ \text{LCM}(3, 4) = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Step 4: Write the intersection set

The intersection \( A \cap B \) will consist of all multiples of the LCM, which is 12. Therefore, we can express the intersection as:
\[ A \cap B = \{ x : x = 12n, n \in \mathbb{Z} \} \]

### Final Answer

Thus, the intersection of sets \( A \) and \( B \) is:
\[ A \cap B = \{ x : x = 12n, n \in \mathbb{Z} \} \]

---