If P is a prime number, then find the LCM of `P^2` and `P^3`.

Options

A. `P^3`
B. P
C. `P^2`
D. `P^4`

Answer

To find the LCM (Least Common Multiple) of \( P^2 \) and \( P^3 \), where \( P \) is a prime number, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Terms**: We have two terms: \( P^2 \) and \( P^3 \).

2. **Express the Terms**:
- \( P^2 \) can be expressed as \( P \times P \).
- \( P^3 \) can be expressed as \( P \times P \times P \).

3. **Determine the Highest Power**:
- The powers of \( P \) in the two terms are 2 (from \( P^2 \)) and 3 (from \( P^3 \)).
- To find the LCM, we take the highest power of \( P \) from both terms.

4. **Calculate the LCM**:
- The highest power of \( P \) between \( P^2 \) and \( P^3 \) is \( P^3 \).
- Therefore, the LCM of \( P^2 \) and \( P^3 \) is \( P^3 \).

### Final Answer:
The LCM of \( P^2 \) and \( P^3 \) is \( P^3 \).

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