The number of positive integral solutions of `x^4-y^4=3789108` is

Options

A. 0
B. 1
C. 2
D. 4

Explanation

To solve the equation \( x^4 - y^4 = 3789108 \) and find the number of positive integral solutions, we can follow these steps:

### Step 1: Recognize the form of the equation
The equation \( x^4 - y^4 \) can be factored using the difference of squares:
\[ x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) \]
Further, we can factor \( x^2 - y^2 \) as:
\[ x^2 - y^2 = (x - y)(x + y) \]
Thus, we can rewrite the equation as:
\[ (x - y)(x + y)(x^2 + y^2) = 3789108 \]

### Step 2: Analyze the parity of \( x \) and \( y \)
Since \( 3789108 \) is an even number (its last digit is 8), both \( x^4 \) and \( y^4 \) must be either both even or both odd. This means:
- If both \( x \) and \( y \) are even, \( x^4 \) and \( y^4 \) are even.
- If both \( x \) and \( y \) are odd, \( x^4 \) and \( y^4 \) are also even.

### Step 3: Check divisibility by 8
Next, we need to check if \( 3789108 \) is divisible by 8. We can do this by performing the division:
\[ 3789108 \div 8 \]
Calculating this, we find:
\[ 3789108 \div 8 = 473638.5 \]
This indicates that \( 3789108 \) is not divisible by 8, as it gives a remainder.

### Step 4: Conclusion based on divisibility
Since \( x^4 - y^4 \) must be divisible by 8 (as shown in the factorization), but \( 3789108 \) is not divisible by 8, we conclude that there are no positive integral solutions for the equation \( x^4 - y^4 = 3789108 \).

### Final Answer
Thus, the number of positive integral solutions of the equation \( x^4 - y^4 = 3789108 \) is:
\[ \boxed{0} \]