If `N_(a) = {ax| x in N}`, then what is `N_(12) cap N_(8)` equal to ?

Options

A. A. `N_(12)`
B. B. `N_(20)`
C. C. `N_(24)`
D. D. `N_(48)`

Answer

To solve the problem, we need to find the intersection of the sets \( N_{12} \) and \( N_{8} \). Let's break down the steps:

### Step 1: Define the Sets
We start by defining the sets based on the given notation:
- \( N_{a} = \{ ax | x \in N \} \)
- Therefore, \( N_{8} = \{ 8x | x \in N \} \) and \( N_{12} = \{ 12x | x \in N \} \)

### Step 2: List the Elements of Each Set
Next, we list the elements of both sets:
- \( N_{8} = \{ 8, 16, 24, 32, 40, 48, 56, 64, \ldots \} \) (multiples of 8)
- \( N_{12} = \{ 12, 24, 36, 48, 60, 72, \ldots \} \) (multiples of 12)

### Step 3: Find the Intersection
Now, we need to find the common elements in both sets:
- The common elements are those numbers that appear in both \( N_{8} \) and \( N_{12} \).
- From our lists, we can see that the common elements are \( 24, 48, 72, \ldots \)

### Step 4: Express the Intersection
The intersection can be expressed as:
- \( N_{12} \cap N_{8} = \{ 24, 48, 72, \ldots \} \)
- This set consists of multiples of 24.

### Step 5: Conclusion
Thus, we can conclude that:
- \( N_{12} \cap N_{8} = N_{24} \)

### Final Answer
The final answer is:
\[ N_{12} \cap N_{8} = N_{24} \]

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