The give expression `n^4 - n^2` is divisible by for `n in I^+` and `n > 2`:

Options

A. 4
B. 8
C. 12
D. all of these

Answer

To determine the divisibility of the expression \( n^4 - n^2 \) for \( n \in \mathbb{I}^+ \) and \( n > 2 \), we can follow these steps:

### Step 1: Factor the expression
The expression \( n^4 - n^2 \) can be factored by taking out the common term \( n^2 \):
\[ n^4 - n^2 = n^2(n^2 - 1) \]
This can be further factored since \( n^2 - 1 \) is a difference of squares:
\[ n^2(n^2 - 1) = n^2(n - 1)(n + 1) \]

### Step 2: Analyze the factors
Now, we have the expression in the form:
\[ n^2(n - 1)(n + 1) \]
Here, \( n^2 \) is always divisible by 1 and \( n^2 \) itself is a perfect square. The terms \( (n - 1) \) and \( (n + 1) \) are two consecutive integers, which means one of them is always even.

### Step 3: Determine divisibility by 4
Since one of \( (n - 1) \) or \( (n + 1) \) is even, and \( n^2 \) is also even when \( n \) is even, we can conclude:
- If \( n \) is even, \( n^2 \) is divisible by 4.
- If \( n \) is odd, \( n^2 \) is odd, but \( (n - 1)(n + 1) \) will be even, and one of those factors will be divisible by 2, making the product \( n^2(n - 1)(n + 1) \) divisible by 4.

### Step 4: Check divisibility by 8
For \( n > 2 \):
- If \( n \) is even, \( n^2 \) is divisible by 4, and since \( (n - 1)(n + 1) \) contains two consecutive integers, one of them is even, thus the product is divisible by 8.
- If \( n \) is odd, \( n^2 \) is odd, but \( (n - 1)(n + 1) \) is even, and one of them is divisible by 4, hence the product is divisible by 8.

### Step 5: Check divisibility by 12
The expression \( n^2(n - 1)(n + 1) \) will also be divisible by 12:
- Since \( n^2 \) contributes at least one factor of 2, and either \( (n - 1) \) or \( (n + 1) \) contributes another factor of 2, we have at least two factors of 2.
- Additionally, one of the terms \( (n - 1) \) or \( (n + 1) \) will also be divisible by 3 when \( n \) is greater than 2.

### Conclusion
Thus, the expression \( n^4 - n^2 \) is divisible by 4, 8, and 12 for all integers \( n > 2 \).

### Final Answer
The expression \( n^4 - n^2 \) is divisible by 4, 8, and 12 for \( n \in \mathbb{I}^+ \) and \( n > 2 \).