If N is the number of positive integral solutions of `x_1x_2x_3x_4 = 770`, then N =

Options

A. N is divisble by 4 distinct primes
B. N is a perfect square
C. N is a perfect fourth power
D. N is a perfect 8th power

Answer

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

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

\[ 770 = 77 \times 10 = 7 \times 11 \times 2 \times 5 \]

Thus, the prime factorization of 770 is:

\[ 770 = 2^1 \times 5^1 \times 7^1 \times 11^1 \]

### Step 2: Distributing the Prime Factors
Next, we need to distribute these prime factors among the 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
To find the number of positive integral solutions, we can use the stars and bars method. Each prime factor can be assigned to any of the four variables.

For each prime factor \( p^k \) (where \( k \) is the exponent), the number of ways to distribute \( k \) identical items (the prime factors) into \( n \) distinct groups (the variables) is given by the formula:

\[ \text{Number of ways} = \binom{n+k-1}{k} \]

In our case, we have four prime factors (2, 5, 7, 11), each raised to the power of 1. Therefore, for each prime factor, we have:

\[ \text{Number of ways for each prime factor} = \binom{4+1-1}{1} = \binom{4}{1} = 4 \]

### Step 4: Total Combinations
Since the distribution of each prime factor is independent, we 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 \( N \) is:

\[ \boxed{256} \]

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