If p, q are the positive integers and r, s, t are prime numbers such that the L.C.M. of p, q is `r^4s^7t^2` then the number of ordered pairs (p, q) is :

Options

A. 567
B. 675
C. 120
D. 180

Answer

To solve the problem, we need to find the number of ordered pairs \((p, q)\) such that the least common multiple (L.C.M.) of \(p\) and \(q\) is given by \(r^4 s^7 t^2\), where \(r\), \(s\), and \(t\) are prime numbers.

### Step-by-Step Solution:

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

2. **Setting up the equations**:
Given that \(\text{L.C.M.}(p, q) = r^4 s^7 t^2\), we have:
\[ \max(a_1, a_2) = 4 \]
\[ \max(b_1, b_2) = 7 \]
\[ \max(c_1, c_2) = 2 \]

3. **Finding possible values for \(a_1\) and \(a_2\)**:
For \(\max(a_1, a_2) = 4\), the possible pairs \((a_1, a_2)\) are:
- (4, 0)
- (4, 1)
- (4, 2)
- (4, 3)
- (4, 4)
- (0, 4)
- (1, 4)
- (2, 4)
- (3, 4)

This gives us a total of 9 combinations.

4. **Finding possible values for \(b_1\) and \(b_2\)**:
For \(\max(b_1, b_2) = 7\), the possible pairs \((b_1, b_2)\) are:
- (7, 0)
- (7, 1)
- (7, 2)
- (7, 3)
- (7, 4)
- (7, 5)
- (7, 6)
- (7, 7)
- (0, 7)
- (1, 7)
- (2, 7)
- (3, 7)
- (4, 7)
- (5, 7)
- (6, 7)

This gives us a total of 15 combinations.

5. **Finding possible values for \(c_1\) and \(c_2\)**:
For \(\max(c_1, c_2) = 2\), the possible pairs \((c_1, c_2)\) are:
- (2, 0)
- (2, 1)
- (2, 2)
- (0, 2)
- (1, 2)

This gives us a total of 5 combinations.

6. **Calculating the total number of ordered pairs**:
The total number of ordered pairs \((p, q)\) is the product of the number of combinations for \(a\), \(b\), and \(c\):
\[ \text{Total pairs} = 9 \times 15 \times 5 = 675 \]

### Final Answer:
The number of ordered pairs \((p, q)\) is \(675\).