lf `r, s, t` are prime numbers and `p, q` are the positive integers such that their LCM of `p,q` is `r^2 t^4 s^2,` then the numbers of ordered pair of `(p, q)` is (A) `252` (B) `254` (C) `225` (D) `224`

Options

A. 252
B. 254
C. 225
D. 224

Explanation

To solve the problem, we need to find the number of ordered pairs \((p, q)\) such that the least common multiple (LCM) of \(p\) and \(q\) is given by \(r^2 t^4 s^2\), where \(r\), \(s\), and \(t\) are prime numbers.

### Step-by-Step Solution:

1. **Understanding the LCM**:
The LCM of two numbers \(p\) and \(q\) can be expressed in terms of their prime factorization. Given that the LCM is \(r^2 t^4 s^2\), we can denote the prime factorization of \(p\) and \(q\) as follows:
\[ p = r^{a_1} t^{b_1} s^{c_1}, \quad q = r^{a_2} t^{b_2} s^{c_2} \]
where \(a_1, a_2, b_1, b_2, c_1, c_2\) are non-negative integers.

2. **Setting Up Conditions for LCM**:
The LCM of \(p\) and \(q\) is given by:
\[ \text{LCM}(p, q) = r^{\max(a_1, a_2)} t^{\max(b_1, b_2)} s^{\max(c_1, c_2)} \]
We need:
\[ \max(a_1, a_2) = 2, \quad \max(b_1, b_2) = 4, \quad \max(c_1, c_2) = 2 \]

3. **Finding Cases for Each Prime Factor**:
- **For \(r^2\)**:
- Possible pairs \((a_1, a_2)\) that satisfy \(\max(a_1, a_2) = 2\):
- \( (2, 0), (2, 1), (2, 2), (0, 2), (1, 2) \)
- This gives us **5 cases**.

- **For \(t^4\)**:
- Possible pairs \((b_1, b_2)\) that satisfy \(\max(b_1, b_2) = 4\):
- \( (4, 0), (4, 1), (4, 2), (4, 3), (4, 4), (0, 4), (1, 4), (2, 4), (3, 4) \)
- This gives us **9 cases**.

- **For \(s^2\)**:
- Possible pairs \((c_1, c_2)\) that satisfy \(\max(c_1, c_2) = 2\):
- \( (2, 0), (2, 1), (2, 2), (0, 2), (1, 2) \)
- This gives us **5 cases**.

4. **Calculating Total Ordered Pairs**:
The total number of ordered pairs \((p, q)\) can be calculated by multiplying the number of cases for each prime factor:
\[ \text{Total pairs} = 5 \times 9 \times 5 = 225 \]

### Final Answer:
The number of ordered pairs \((p, q)\) is **225**.