The LCM of 12, 15, 20, 27 is .

Options

A. 270
B. 360
C. 480
D. 540

Answer

To find the LCM (Least Common Multiple) of the numbers 12, 15, 20, and 27, we can use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **12**:
- 12 = 2 × 2 × 3 = \(2^2 \times 3^1\)

- **15**:
- 15 = 3 × 5 = \(3^1 \times 5^1\)

- **20**:
- 20 = 2 × 2 × 5 = \(2^2 \times 5^1\)

- **27**:
- 27 = 3 × 3 × 3 = \(3^3\)

### Step 2: Identify the Highest Powers
Next, we identify the highest power of each prime factor from all the factorizations:

- For **2**: The highest power is \(2^2\) (from 12 and 20).
- For **3**: The highest power is \(3^3\) (from 27).
- For **5**: The highest power is \(5^1\) (from 15 and 20).

### Step 3: Multiply the Highest Powers
Now, we multiply these highest powers together to find the LCM:

\[ LCM = 2^2 \times 3^3 \times 5^1 \]

Calculating this step-by-step:

1. Calculate \(2^2 = 4\)
2. Calculate \(3^3 = 27\)
3. Calculate \(5^1 = 5\)

Now multiply these results together:

\[ LCM = 4 \times 27 \times 5 \]

### Step 4: Perform the Multiplication
First, multiply \(4\) and \(27\):

\[ 4 \times 27 = 108 \]

Now multiply \(108\) by \(5\):

\[ 108 \times 5 = 540 \]

### Final Answer
Thus, the LCM of 12, 15, 20, and 27 is **540**.

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