The number of positive integral pairs (x, y) satisfying the equation ` x^(2) - y^(2) = 3370 ` is :

Options

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

Answer

To solve the equation \( x^2 - y^2 = 3370 \) for positive integral pairs \( (x, y) \), we can start by factoring the left-hand side of the equation.

### Step 1: Factor the equation
The equation \( x^2 - y^2 \) can be factored using the difference of squares:
\[ x^2 - y^2 = (x - y)(x + y) \]
Thus, we can rewrite the equation as:
\[ (x - y)(x + y) = 3370 \]

### Step 2: Find the factors of 3370
Next, we need to find the positive factors of 3370. We can start by determining the prime factorization of 3370:
- 3370 is even, so we divide by 2:
\[ 3370 \div 2 = 1685 \]
- Next, we check 1685 for divisibility by 5 (it ends in 5):
\[ 1685 \div 5 = 337 \]
- Now, we check if 337 is prime. It is not divisible by 2, 3, 5, or 7. However, it is divisible by 11:
\[ 337 \div 11 = 31 \]
Thus, the prime factorization of 3370 is:
\[ 3370 = 2 \times 5 \times 11 \times 31 \]

### Step 3: List the factor pairs
Now we can list the factor pairs of 3370:
- \( (1, 3370) \)
- \( (2, 1685) \)
- \( (5, 674) \)
- \( (10, 337) \)
- \( (11, 307) \)
- \( (31, 110) \)
- \( (62, 55) \)

### Step 4: Set up equations for each factor pair
For each factor pair \( (a, b) \) where \( a = x - y \) and \( b = x + y \), we can set up the following equations:
1. \( x - y = a \)
2. \( x + y = b \)

We can solve for \( x \) and \( y \):
\[ x = \frac{(x - y) + (x + y)}{2} = \frac{a + b}{2} \]
\[ y = \frac{(x + y) - (x - y)}{2} = \frac{b - a}{2} \]

### Step 5: Check for positive integers
For \( x \) and \( y \) to be positive integers, both \( a + b \) and \( b - a \) must be even numbers. This means \( a \) and \( b \) must have the same parity (both even or both odd).

### Step 6: Analyze each factor pair
Now we check each factor pair:
1. \( (1, 3370) \): \( a + b = 3371 \) (odd), \( b - a = 3369 \) (odd) → Not valid
2. \( (2, 1685) \): \( a + b = 1687 \) (odd), \( b - a = 1683 \) (odd) → Not valid
3. \( (5, 674) \): \( a + b = 679 \) (odd), \( b - a = 669 \) (odd) → Not valid
4. \( (10, 337) \): \( a + b = 347 \) (odd), \( b - a = 327 \) (odd) → Not valid
5. \( (11, 307) \): \( a + b = 318 \) (even), \( b - a = 296 \) (even) → Valid
6. \( (31, 110) \): \( a + b = 141 \) (odd), \( b - a = 79 \) (odd) → Not valid
7. \( (62, 55) \): \( a + b = 117 \) (odd), \( b - a = -7 \) (not positive) → Not valid

### Step 7: Count valid pairs
From our analysis, only the factor pair \( (11, 307) \) gives us valid positive integers for \( x \) and \( y \).

### Conclusion
Thus, the number of positive integral pairs \( (x, y) \) satisfying the equation \( x^2 - y^2 = 3370 \) is **1**.