The sum of first three common multiples of 3, 4 and 9 is

Options

A. 108
B. 144
C. 252
D. 216

Answer

To find the sum of the first three common multiples of 3, 4, and 9, we will follow these steps:

### Step 1: Find the LCM of 3, 4, and 9
To find the least common multiple (LCM), we can use the prime factorization method:

- **3** is a prime number: \(3 = 3^1\)
- **4** can be factored as \(2^2\)
- **9** can be factored as \(3^2\)

Now, we take the highest power of each prime number:
- For prime number **2**, the highest power is \(2^2\) (from 4).
- For prime number **3**, the highest power is \(3^2\) (from 9).

Thus, the LCM can be calculated as:
\[ \text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 2: Identify the first three common multiples
The common multiples of 3, 4, and 9 are multiples of the LCM we just found. Therefore, the first three common multiples are:
1. \(36 \times 1 = 36\)
2. \(36 \times 2 = 72\)
3. \(36 \times 3 = 108\)

### Step 3: Calculate the sum of the first three common multiples
Now, we add these three common multiples together:
\[ 36 + 72 + 108 \]

Calculating step by step:
1. First, add \(36 + 72\):
\[ 36 + 72 = 108 \]
2. Next, add the result to \(108\):
\[ 108 + 108 = 216 \]

### Final Answer
The sum of the first three common multiples of 3, 4, and 9 is \(216\).

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