To find the number of ordered pairs of positive integers \((a, b)\) such that their Least Common Multiple (LCM) is \(7^2 \times 11^3 \times 19^4\), we can follow these steps:
### Step 1: Express \(a\) and \(b\) in terms of their prime factors
Assume:
\[
a = 7^{x_1} \times 11^{y_1} \times 19^{z_1}
\]
\[
b = 7^{x_2} \times 11^{y_2} \times 19^{z_2}
\]
### Step 2: Set up the conditions for LCM
The LCM of \(a\) and \(b\) 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 \(7^2 \times 11^3 \times 19^4\). Therefore, we have the following conditions:
1. \(\max(x_1, x_2) = 2\)
2. \(\max(y_1, y_2) = 3\)
3. \(\max(z_1, z_2) = 4\)
### Step 3: Analyze each condition separately
#### For \(x_1\) and \(x_2\):
- The values can be \(0, 1, 2\).
- To satisfy \(\max(x_1, x_2) = 2\), we can have:
- \( (2, 0) \)
- \( (0, 2) \)
- \( (2, 1) \)
- \( (1, 2) \)
- \( (2, 2) \)
Thus, there are **5 valid pairs** for \((x_1, x_2)\).
#### For \(y_1\) and \(y_2\):
- The values can be \(0, 1, 2, 3\).
- To satisfy \(\max(y_1, y_2) = 3\), we can have:
- \( (3, 0) \)
- \( (0, 3) \)
- \( (3, 1) \)
- \( (1, 3) \)
- \( (3, 2) \)
- \( (2, 3) \)
- \( (3, 3) \)
Thus, there are **7 valid pairs** for \((y_1, y_2)\).
#### For \(z_1\) and \(z_2\):
- The values can be \(0, 1, 2, 3, 4\).
- To satisfy \(\max(z_1, z_2) = 4\), we can have:
- \( (4, 0) \)
- \( (0, 4) \)
- \( (4, 1) \)
- \( (1, 4) \)
- \( (4, 2) \)
- \( (2, 4) \)
- \( (4, 3) \)
- \( (3, 4) \)
- \( (4, 4) \)
Thus, there are **9 valid pairs** for \((z_1, z_2)\).
### Step 4: Calculate the total number of ordered pairs
The total number of ordered pairs \((a, b)\) is the product of the number of valid pairs for each prime factor:
\[
\text{Total pairs} = 5 \times 7 \times 9 = 315
\]
### Final Answer
The number of ordered pairs of positive integers \((a, b)\) such that their LCM is \(7^2 \times 11^3 \times 19^4\) is **315**.
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