If N is the number of positive integral solutions of the equation `x_(1)x_(2)x_(3)x_(4)=770`, then the value of N is

Options

A. 250
B. 252
C. 254
D. 256

Answer

To find the number of positive integral solutions of the equation \( x_1 x_2 x_3 x_4 = 770 \), we can follow these steps:

### Step 1: Prime Factorization of 770
First, we need to factor the number 770 into its prime factors.

\[ 770 = 2 \times 5 \times 7 \times 11 \]

### Step 2: Distributing Prime Factors
Next, we need to distribute these prime factors among the four variables \( x_1, x_2, x_3, \) and \( x_4 \). Each variable can take any combination of the prime factors.

### Step 3: Using the Stars and Bars Method
We can represent the distribution of each prime factor among the four variables. For each prime factor, we can think of it as distributing identical objects (the prime factors) into distinct boxes (the variables).

For each prime factor, we can use the "stars and bars" theorem. The number of ways to distribute \( k \) identical items into \( n \) distinct boxes is given by the formula:

\[ \binom{n + k - 1}{k} \]

In our case, we have 4 variables (boxes) and we need to distribute 1 of each prime factor (identical items) among them.

### Step 4: Calculating for Each Prime Factor
Since we have four distinct prime factors (2, 5, 7, and 11), we will calculate the distribution for each:

- For the prime factor 2: The number of ways to distribute it is \( \binom{4 - 1}{1} = 4 \).
- For the prime factor 5: The number of ways to distribute it is \( \binom{4 - 1}{1} = 4 \).
- For the prime factor 7: The number of ways to distribute it is \( \binom{4 - 1}{1} = 4 \).
- For the prime factor 11: The number of ways to distribute it is \( \binom{4 - 1}{1} = 4 \).

### Step 5: Total Combinations
Now, since the distributions are independent, we can multiply the number of ways for each prime factor:

\[ N = 4 \times 4 \times 4 \times 4 = 4^4 = 256 \]

### Conclusion
Thus, the number of positive integral solutions \( N \) of the equation \( x_1 x_2 x_3 x_4 = 770 \) is:

\[ \boxed{256} \]