If `n` is the number of positive integral solution of `x_(1), x_(2)x_(3)x_(4) = 210`, then which of the following is incorrect ?

Options

A. n must be a perfect square
B. n must be a perfect `4^(th)` power
C. n must be a perfact `8^(th)` power
D. n must be divisible by an odd prime number

Explanation

To solve the problem of finding the number of positive integral solutions for the equation \( x_1 x_2 x_3 x_4 = 210 \), we will follow these steps:

### Step 1: Prime Factorization of 210
First, we need to factor 210 into its prime factors.
\[ 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \]

### Step 2: Using the Formula for Positive Integral Solutions
The number of positive integral solutions of the equation \( x_1 x_2 x_3 x_4 = N \) can be calculated using the formula:
\[ n = (e_1 + k - 1)(e_2 + k - 1)(e_3 + k - 1) \ldots (e_m + k - 1) \]
where \( e_i \) are the exponents in the prime factorization of \( N \) and \( k \) is the number of variables (in this case, \( k = 4 \)).

### Step 3: Applying the Formula
For our case, we have:
- \( e_1 = 1 \) (for prime 2)
- \( e_2 = 1 \) (for prime 3)
- \( e_3 = 1 \) (for prime 5)
- \( e_4 = 1 \) (for prime 7)

Thus, we can calculate:
\[ n = (1 + 4 - 1)(1 + 4 - 1)(1 + 4 - 1)(1 + 4 - 1) = 4 \times 4 \times 4 \times 4 = 4^4 \]

### Step 4: Calculate \( 4^4 \)
Now, we compute \( 4^4 \):
\[ 4^4 = 256 \]

### Step 5: Analyzing the Options
Now that we have \( n = 256 \), we need to analyze the given options to determine which one is incorrect:
1. \( n \) is a perfect square.
2. \( n \) must be a perfect fourth power.
3. \( n \) must be a perfect eighth power.
4. \( n \) must be divisible by a prime number.

### Step 6: Evaluating Each Statement
1. **Perfect Square**: \( 256 = 16^2 \) (True)
2. **Perfect Fourth Power**: \( 256 = 4^4 \) (True)
3. **Perfect Eighth Power**: \( 256 = 2^8 \) (True)
4. **Divisible by a Prime Number**: \( 256 \) is divisible by \( 2 \) (True)

### Conclusion
All statements are true, so we need to identify which one is incorrect based on the context of the question. The statement that \( n \) must be divisible by a prime number is trivially true for all integers greater than 1. Thus, the statement that is typically considered incorrect in this context is the one that suggests a specific condition that doesn't apply universally.

### Final Answer
The incorrect statement is that "n must be divisible by a prime number" since all integers greater than 1 are divisible by at least one prime number.