If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is `r^(2)s^(4)t^(2)`, then the number of ordered pairs (p,q) is:

Options

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

Answer

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 as \( r^2 s^4 t^2 \), where \( r, s, t \) are prime numbers, and \( p, q \) are positive integers.

### Step-by-Step Solution:

1. **Understanding LCM and Prime Factorization**:
The LCM of two numbers p and q can be expressed in terms of their prime factorizations. If \( p \) and \( q \) have the prime factorization:
\[ p = r^{a_1} s^{b_1} t^{c_1}, \quad q = r^{a_2} s^{b_2} t^{c_2} \]
then the LCM is given by:
\[ \text{LCM}(p, q) = r^{\max(a_1, a_2)} s^{\max(b_1, b_2)} t^{\max(c_1, c_2)} \]

2. **Setting Up the Equations**:
Given that:
\[ \text{LCM}(p, q) = r^2 s^4 t^2 \]
we can set up the following equations based on the exponents:
- \( \max(a_1, a_2) = 2 \)
- \( \max(b_1, b_2) = 4 \)
- \( \max(c_1, c_2) = 2 \)

3. **Finding Possible Values for \( r \)**:
For \( \max(a_1, a_2) = 2 \):
- Possible pairs \((a_1, a_2)\) can be:
- (2, 0)
- (0, 2)
- (1, 1)
- (2, 1)
- (1, 2)
- This gives us 5 combinations.

4. **Finding Possible Values for \( s \)**:
For \( \max(b_1, b_2) = 4 \):
- Possible pairs \((b_1, b_2)\) can be:
- (4, 0)
- (0, 4)
- (1, 4)
- (4, 1)
- (2, 4)
- (4, 2)
- (3, 4)
- (4, 3)
- (4, 4)
- This gives us 9 combinations.

5. **Finding Possible Values for \( t \)**:
For \( \max(c_1, c_2) = 2 \):
- Possible pairs \((c_1, c_2)\) can be:
- (2, 0)
- (0, 2)
- (1, 1)
- (2, 1)
- (1, 2)
- This gives us 5 combinations.

6. **Calculating Total Ordered Pairs**:
The total number of ordered pairs \((p, q)\) is the product of the number of combinations for each prime factor:
\[ \text{Total pairs} = 5 \times 9 \times 5 = 225 \]

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