Find the LCM of 12,15,27 and 30.

Options

A. 600
B. 540
C. 1800
D. 300

Answer

To find the LCM (Least Common Multiple) of the numbers 12, 15, 27, and 30, we will follow these steps:

### Step 1: Find the Prime Factorization of Each Number
- **12**: The prime factorization of 12 is \(2^2 \times 3^1\).
- **15**: The prime factorization of 15 is \(3^1 \times 5^1\).
- **27**: The prime factorization of 27 is \(3^3\).
- **30**: The prime factorization of 30 is \(2^1 \times 3^1 \times 5^1\).

### Step 2: Identify the Highest Power of Each Prime Factor
Now, we will identify the highest power of each prime factor that appears in the factorizations:
- For the prime factor **2**: The highest power is \(2^2\) (from 12).
- For the prime factor **3**: The highest power is \(3^3\) (from 27).
- For the prime factor **5**: The highest power is \(5^1\) (from 15 and 30).

### Step 3: Multiply the Highest Powers Together
Now, we will multiply these highest powers to find the LCM:
\[ LCM = 2^2 \times 3^3 \times 5^1 \]

Calculating this step-by-step:
- First, calculate \(2^2 = 4\).
- Next, calculate \(3^3 = 27\).
- Finally, calculate \(5^1 = 5\).

Now, multiply these results together:
\[ LCM = 4 \times 27 \times 5 \]

### Step 4: Perform the Multiplication
- Calculate \(4 \times 27 = 108\).
- Now, calculate \(108 \times 5 = 540\).

### Conclusion
Thus, the LCM of 12, 15, 27, and 30 is **540**.

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