How many positive integer divides the expression $n(n^2-1)(n^2+3)(n^2+5)$ for every possible value of positive integer $n$?

$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$
To determine how many positive integers divide the expression \( n(n^2-1)(n^2+3)(n^2+5) \) for every possible value of positive integer \( n \), we need to analyze the factors of the expression.

1. **Factorize the expression:**
   \[
   n(n^2-1)(n^2+3)(n^2+5)
   \]
   Notice that \( n^2 - 1 = (n-1)(n+1) \). Therefore, the expression can be rewritten as:
   \[
   n(n-1)(n+1)(n^2+3)(n^2+5)
   \]

2. **Check for specific values of \( n \):**
   Let's start with \( n = 2 \):
   \[
   2(2^2-1)(2^2+3)(2^2+5) = 2 \cdot 3 \cdot 7 \cdot 9 = 2 \cdot 3 \cdot 3 \cdot 3 \cdot 7 = 2 \cdot 3^3 \cdot 7
   \]
   The prime factorization of this product is \( 2 \cdot 3^3 \cdot 7 \).

3. **Determine the divisors:**
   The number of divisors of a number is given by multiplying the incremented exponents of its prime factorization. For \( 2 \cdot 3^3 \cdot 7 \):
   \[
   (1+1)(3+1)(1+1) = 2 \cdot 4 \cdot 2 = 16
   \]
   This means there are 16 divisors of \( 2 \cdot 3^3 \cdot 7 \).

4. **Verify for other values of \( n \):**
   We need to ensure that these divisors are valid for every positive integer \( n \). Consider the general form:
   \[
   n(n-1)(n+1)(n^2+3)(n^2+5)
   \]
   For any \( n \), the expression will always include the factors \( n, n-1, \) and \( n+1 \), which are three consecutive integers. Among any three consecutive integers, at least one is divisible by 2, and at least one is divisible by 3. Additionally, the terms \( n^2+3 \) and \( n^2+5 \) will contribute additional factors, but the key is that the product will always be divisible by the prime factors 2, 3, and 7.

5. **Conclusion:**
   Since the expression \( n(n^2-1)(n^2+3)(n^2+5) \) for any \( n \) will always have the prime factors 2, 3, and 7, the number of positive integer divisors that divide the expression for every \( n \) is 16.

The final answer is \(\boxed{16}\)