Find the number of integral solutions of the equation `x_1x_2x_3x_4 = 462.`

Options

A. A) 2048
B. B) 256
C. C) 24
D. D) 1024

Answer

To find the number of integral solutions of the equation \( x_1 x_2 x_3 x_4 = 462 \), we can follow these steps:

### Step 1: Prime Factorization of 462
First, we need to factor the number 462 into its prime factors.

\[ 462 = 2 \times 3 \times 7 \times 11 \]

### Step 2: Assigning Prime Factors
We have four prime factors: \( 2, 3, 7, \) and \( 11 \). We need to distribute these prime factors among the four variables \( x_1, x_2, x_3, \) and \( x_4 \).

### Step 3: Counting Assignments
Each prime factor can be assigned to any of the four variables. Therefore, for each prime factor, there are 4 choices (it can go to \( x_1, x_2, x_3, \) or \( x_4 \)).

Since there are 4 prime factors, the total number of ways to assign these factors is:

\[ 4^4 = 256 \]

### Step 4: Considering Integral Solutions
Now, we need to consider the integral solutions. The product \( x_1 x_2 x_3 x_4 \) must be positive. This can happen in the following cases:
1. All four variables are positive.
2. Two variables are positive and two are negative.
3. All four variables are negative.

### Step 5: Counting the Cases
We can calculate the number of ways to choose the signs for the variables:

- **Case 1**: All variables are positive. There is only 1 way to choose this case.

- **Case 2**: Choose 2 out of 4 variables to be positive. The number of ways to choose 2 from 4 is given by the binomial coefficient \( \binom{4}{2} \):

\[ \binom{4}{2} = 6 \]

- **Case 3**: All variables are negative. There is only 1 way to choose this case.

Adding these cases together gives:

\[ 1 + 6 + 1 = 8 \]

### Step 6: Total Integral Solutions
Finally, we multiply the number of ways to assign the prime factors by the number of ways to choose the signs:

\[ \text{Total Integral Solutions} = 256 \times 8 = 2048 \]

Thus, the number of integral solutions of the equation \( x_1 x_2 x_3 x_4 = 462 \) is **2048**.

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