If A= {x: x is a multiple of 5} and B] {x: x is a multiple of 9}. Then, `A nn B` is given by

Options

A. `{3, 6, 9,...}`
B. `{15,30,45,...}`
C. `{5,10,15,20,...}`
D. None of these

Answer

To solve the problem of finding the intersection of sets A and B, where:

- \( A = \{ x : x \text{ is a multiple of } 5 \} \)
- \( B = \{ x : x \text{ is a multiple of } 9 \} \)

we will follow these steps:

### Step 1: Define the sets A and B
The set \( A \) consists of all multiples of 5, which can be expressed as:
\[ A = \{ 5n : n \in \mathbb{Z} \} \]
This means \( A \) includes numbers like \( \ldots, -10, -5, 0, 5, 10, 15, 20, \ldots \).

The set \( B \) consists of all multiples of 9, which can be expressed as:
\[ B = \{ 9m : m \in \mathbb{Z} \} \]
This means \( B \) includes numbers like \( \ldots, -27, -18, -9, 0, 9, 18, 27, \ldots \).

### Step 2: Find the intersection \( A \cap B \)
The intersection \( A \cap B \) consists of all elements that are in both sets A and B. To find the common multiples, we need to find the least common multiple (LCM) of 5 and 9.

The prime factorization of 5 is \( 5^1 \) and for 9 it is \( 3^2 \). Since 5 and 9 have no common factors, the LCM is:
\[ \text{LCM}(5, 9) = 5^1 \times 3^2 = 45 \]

### Step 3: Express the intersection
Thus, the intersection \( A \cap B \) can be expressed as:
\[ A \cap B = \{ 45k : k \in \mathbb{Z} \} \]
This means \( A \cap B \) includes numbers like \( \ldots, -90, -45, 0, 45, 90, \ldots \).

### Final Answer
The intersection \( A \cap B \) is:
\[ A \cap B = \{ 45k : k \in \mathbb{Z} \} \]