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 of \( P^2 \) and \( P^3 \) where \( P \) is a prime number, we can follow these steps:

### Step 1: Understand the definitions
- **LCM (Least Common Multiple)**: The LCM of two numbers is the smallest number that is a multiple of both.
- **Prime Number**: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

### Step 2: Write the expressions for \( P^2 \) and \( P^3 \)
- Here, \( P^2 = P \times P \)
- And \( P^3 = P \times P \times P \)

### Step 3: Identify the highest power of \( P \)
- The LCM takes the highest power of each prime factor present in the numbers.
- For \( P^2 \), the highest power of \( P \) is \( P^2 \).
- For \( P^3 \), the highest power of \( P \) is \( P^3 \).

### Step 4: Determine the LCM
- Since we are looking for the highest power of \( P \) between \( P^2 \) and \( P^3 \), we take \( P^3 \) because it is greater than \( P^2 \).
- Therefore, the LCM of \( P^2 \) and \( P^3 \) is \( P^3 \).

### Final Answer
\[ \text{LCM}(P^2, P^3) = P^3 \]

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