P and Q are two positive integers such that `P = p^(3)q`? and Q =` (pq)^(2)`, where p and q are prime numbers.
What is LCM(P, Q)?

Options

A. pq
B. `p^(2) q^(2)`
C. `p^(3) q^(2)`
D. `p^(5) q^(3)`

Answer

To find the LCM (Least Common Multiple) of the two positive integers \( P \) and \( Q \), we start with their definitions:

1. **Given Definitions**:
- \( P = p^3 q \)
- \( Q = (pq)^2 = p^2 q^2 \)

2. **Prime Factorization**:
- The prime factorization of \( P \) is:
\[ P = p^3 \cdot q^1 \]
- The prime factorization of \( Q \) is:
\[ Q = p^2 \cdot q^2 \]

3. **Finding the LCM**:
- The LCM of two numbers is found by taking the highest power of each prime factor that appears in their factorizations.
- For the prime \( p \):
- In \( P \), the power is \( 3 \) (from \( p^3 \)).
- In \( Q \), the power is \( 2 \) (from \( p^2 \)).
- The highest power is \( 3 \).
- For the prime \( q \):
- In \( P \), the power is \( 1 \) (from \( q^1 \)).
- In \( Q \), the power is \( 2 \) (from \( q^2 \)).
- The highest power is \( 2 \).

4. **Combining the Highest Powers**:
- Therefore, the LCM of \( P \) and \( Q \) is:
\[ \text{LCM}(P, Q) = p^3 \cdot q^2 \]

5. **Final Answer**:
- Thus, the LCM of \( P \) and \( Q \) is:
\[ \text{LCM}(P, Q) = p^3 q^2 \]