The number of ordered pairs of positive integers (a, b), such that their Least Common Multiple is the given positive integer `7^(2)xx11^(3)xx19^(4)`, is equal to

Answer

To find the number of ordered pairs of positive integers \( (a, b) \) such that their Least Common Multiple (LCM) is given by \( 7^2 \times 11^3 \times 19^4 \), we can follow these steps:

### Step 1: Understand the LCM and its properties
The LCM of two numbers \( a \) and \( b \) can be expressed in terms of their prime factorization. If \( a \) and \( b \) have the prime factorization:
- \( a = 7^{x_1} \times 11^{y_1} \times 19^{z_1} \)
- \( b = 7^{x_2} \times 11^{y_2} \times 19^{z_2} \)

Then, the LCM can be expressed as:
\[ \text{LCM}(a, b) = 7^{\max(x_1, x_2)} \times 11^{\max(y_1, y_2)} \times 19^{\max(z_1, z_2)} \]
We want this to equal \( 7^2 \times 11^3 \times 19^4 \).

### Step 2: Determine the constraints for each prime factor
For each prime factor, we need to find the possible values of \( x_1, x_2 \), \( y_1, y_2 \), and \( z_1, z_2 \) such that:
- \( \max(x_1, x_2) = 2 \)
- \( \max(y_1, y_2) = 3 \)
- \( \max(z_1, z_2) = 4 \)

### Step 3: Analyze each prime factor

1. **For \( 7^2 \)**:
- Possible pairs \( (x_1, x_2) \) such that \( \max(x_1, x_2) = 2 \):
- \( (2, 0) \)
- \( (2, 1) \)
- \( (2, 2) \)
- \( (0, 2) \)
- \( (1, 2) \)
- Total combinations: 5 ways.

2. **For \( 11^3 \)**:
- Possible pairs \( (y_1, y_2) \) such that \( \max(y_1, y_2) = 3 \):
- \( (3, 0) \)
- \( (3, 1) \)
- \( (3, 2) \)
- \( (3, 3) \)
- \( (0, 3) \)
- \( (1, 3) \)
- \( (2, 3) \)
- Total combinations: 7 ways.

3. **For \( 19^4 \)**:
- Possible pairs \( (z_1, z_2) \) such that \( \max(z_1, z_2) = 4 \):
- \( (4, 0) \)
- \( (4, 1) \)
- \( (4, 2) \)
- \( (4, 3) \)
- \( (4, 4) \)
- \( (0, 4) \)
- \( (1, 4) \)
- \( (2, 4) \)
- \( (3, 4) \)
- Total combinations: 9 ways.

### Step 4: Calculate the total number of ordered pairs
To find the total number of ordered pairs \( (a, b) \), we multiply the number of combinations for each prime factor:
\[ \text{Total pairs} = 5 \times 7 \times 9 = 315 \]

### Final Answer
The number of ordered pairs \( (a, b) \) such that their LCM is \( 7^2 \times 11^3 \times 19^4 \) is \( \boxed{315} \).