Write the first three common multiples of `10` and `15`

Options

A. `60, 90, 120`
B. `30, 45, 60`
C. `15, 60, 90`
D. `30, 60, 90`

Answer

To find the first three common multiples of 10 and 15, we can follow these steps:

### Step 1: Find the LCM (Least Common Multiple) of 10 and 15
To find the LCM, we can use the prime factorization method.

- The prime factorization of 10 is:
- \(10 = 2 \times 5\)

- The prime factorization of 15 is:
- \(15 = 3 \times 5\)

### Step 2: Identify the highest powers of all prime factors
From the factorizations, we take the highest power of each prime number:
- For \(2\), the highest power is \(2^1\) (from 10).
- For \(3\), the highest power is \(3^1\) (from 15).
- For \(5\), the highest power is \(5^1\) (common in both).

### Step 3: Calculate the LCM
Now we multiply these together:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30 \]

### Step 4: List the first three common multiples
The first common multiple is the LCM itself, which is 30. The next common multiples can be found by multiplying the LCM by 2 and 3:
- First common multiple: \(30\)
- Second common multiple: \(30 \times 2 = 60\)
- Third common multiple: \(30 \times 3 = 90\)

### Final Answer
The first three common multiples of 10 and 15 are:
- **30, 60, 90**

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