Statement-1: If N the number of positive integral solutions of `x_(1)x_(2)x_(3)x_(4)=770`, then N is divisible by 4 distinct prime numbers.
Statement-2: Prime numbers are 2,3,5,7,11,13, . .

Options

A. Statement-1 is true, statement-2 is true, statement-2 is a correct explanation for statement-1
B. Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-1
C. Statement-1 is true, statement-2 is false
D. Statement-1 is false, statement-2 is true

Explanation

To solve the problem of finding the number of positive integral solutions to 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 770 into its prime factors.

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

### Step 2: Number of Positive Integral Solutions
The equation \( x_1 x_2 x_3 x_4 = 770 \) can be rephrased in terms of distributing the prime factors among the variables \( x_1, x_2, x_3, \) and \( x_4 \). Each \( x_i \) can be expressed in terms 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" theorem. Each prime factor can be assigned to any of the four variables. Since we have four distinct prime factors (2, 5, 7, and 11), we can assign each factor to any of the four variables.

The number of ways to assign four distinct items (the prime factors) to four distinct groups (the variables) is given by:

\[ 4^4 \]

Calculating this gives:

\[ 4^4 = 256 \]

### Step 4: Checking Divisibility by Distinct Prime Numbers
Now, we need to check if 256 is divisible by four distinct prime numbers. The prime factorization of 256 is:

\[ 256 = 2^8 \]

This shows that 256 is only divisible by the prime number 2. It is not divisible by 5, 7, or 11, which are the other prime factors of 770.

### Conclusion
Since 256 is not divisible by four distinct prime numbers, we conclude that the first statement is false.

### Final Verification of the Second Statement
The second statement lists the prime numbers as 2, 3, 5, 7, 11, 13, etc. This statement is true as these numbers are indeed prime.

### Summary of Statements
- Statement 1: False (N is not divisible by four distinct prime numbers)
- Statement 2: True (The listed numbers are prime)

### Final Answer
The correct answer is that Statement 1 is false and Statement 2 is true.

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