If a N = `{ ax : x in N}` then the set `4N cap 6N` is

Options

A. 8N
B. 10 N
C. 12 N
D. Non of these

Explanation

To solve the problem, we need to find the intersection of the sets \(4N\) and \(6N\), where \(N\) is defined as \(N = \{ ax : x \in N \}\). This means that \(N\) consists of multiples of \(a\).

### Step-by-step Solution:

1. **Understanding the Sets**:
- The set \(N\) consists of all multiples of \(a\):
\[ N = \{ a, 2a, 3a, \ldots \} \]
- The set \(4N\) consists of all multiples of \(4\) that are also multiples of \(a\):
\[ 4N = \{ 4a, 8a, 12a, \ldots \} \]
- The set \(6N\) consists of all multiples of \(6\) that are also multiples of \(a\):
\[ 6N = \{ 6a, 12a, 18a, \ldots \} \]

2. **Finding the Intersection**:
- We need to find the intersection \(4N \cap 6N\).
- This intersection will consist of all elements that are common to both sets \(4N\) and \(6N\).

3. **Finding Common Multiples**:
- To find the common elements, we look for the least common multiple (LCM) of \(4\) and \(6\):
\[ \text{LCM}(4, 6) = 12 \]
- Therefore, the common multiples of \(4\) and \(6\) will be multiples of \(12\).

4. **Expressing the Intersection**:
- The intersection \(4N \cap 6N\) will consist of all multiples of \(12\) that are also multiples of \(a\):
\[ 4N \cap 6N = \{ 12a, 24a, 36a, \ldots \} = 12N \]

5. **Final Answer**:
- Thus, the intersection \(4N \cap 6N\) is:
\[ 4N \cap 6N = 12N \]

### Conclusion:
The set \(4N \cap 6N\) is equal to \(12N\).