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

Options

A. 128
B. 1024
C. 256
D. 64

Answer

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

### Step 1: Factor the number 462
First, we need to find the prime factorization of 462.

- Divide by 2:
\[ 462 \div 2 = 231 \]
- Divide by 3:
\[ 231 \div 3 = 77 \]
- Divide by 7:
\[ 77 \div 7 = 11 \]
- 11 is a prime number.

Thus, the prime factorization of 462 is:
\[ 462 = 2^1 \times 3^1 \times 7^1 \times 11^1 \]

### Step 2: Distribute the prime factors among the variables
We have four variables \( x_1, x_2, x_3, x_4 \) and we need to distribute the prime factors \( 2^1, 3^1, 7^1, 11^1 \) among these variables.

### Step 3: Assign the prime factors
Each prime factor can be assigned to any of the four variables. Since we have four different prime factors, we can assign each factor to any of the four variables independently.

### Step 4: Calculate the number of distributions
For each of the 4 prime factors, we have 4 choices (it can go to \( x_1, x_2, x_3, \) or \( x_4 \)). Therefore, the total number of ways to assign the prime factors is:
\[ 4 \times 4 \times 4 \times 4 = 4^4 \]

### Step 5: Compute \( 4^4 \)
Now we calculate \( 4^4 \):
\[ 4^4 = 256 \]

### Conclusion
Thus, the total number of positive integral solutions of the equation \( x_1 x_2 x_3 x_4 = 462 \) is \( 256 \).