Find the LCM of 12 and 15 by prime factorization method.

Options

A. `30`
B. `40`
C. `50`
D. `60`

Explanation

To find the LCM (Least Common Multiple) of 12 and 15 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 12
- Start by dividing 12 by the smallest prime number, which is 2.
- \( 12 \div 2 = 6 \)
- Divide 6 by 2 again.
- \( 6 \div 2 = 3 \)
- Now, 3 is a prime number, so we stop here.
- The prime factorization of 12 is:
- \( 12 = 2^2 \times 3^1 \)

### Step 2: Prime Factorization of 15
- Start by dividing 15 by the smallest prime number, which is 3.
- \( 15 \div 3 = 5 \)
- Now, 5 is a prime number, so we stop here.
- The prime factorization of 15 is:
- \( 15 = 3^1 \times 5^1 \)

### Step 3: Identify the Highest Powers of Each Prime Factor
- From the prime factorizations:
- For 2: The highest power is \( 2^2 \) (from 12).
- For 3: The highest power is \( 3^1 \) (common in both).
- For 5: The highest power is \( 5^1 \) (from 15).

### Step 4: Calculate the LCM
- Now, multiply the highest powers of all prime factors together:
- \( LCM = 2^2 \times 3^1 \times 5^1 \)
- \( LCM = 4 \times 3 \times 5 \)
- First, calculate \( 4 \times 3 = 12 \).
- Then, calculate \( 12 \times 5 = 60 \).

### Final Answer
- The LCM of 12 and 15 is **60**.

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