Find the LCM of `2 xx 3 xx 5 xx 7` and `3 xx 5 xx 7 xx 11`

Options

A. `2 xx 3 xx 5 xx 7 xx 11`
B. `2 xx 3^2 xx 3^2 xx 7^2 xx 11`
C. `3 xx 5 xx 7`
D. None of the above

Answer

To find the LCM (Least Common Multiple) of the two numbers \(2 \times 3 \times 5 \times 7\) and \(3 \times 5 \times 7 \times 11\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the prime factorization of each number**:
- The first number is \(2 \times 3 \times 5 \times 7\).
- The second number is \(3 \times 5 \times 7 \times 11\).

2. **List all unique prime factors**:
- From the first number, the prime factors are: \(2, 3, 5, 7\).
- From the second number, the prime factors are: \(3, 5, 7, 11\).
- Combining these, the unique prime factors are: \(2, 3, 5, 7, 11\).

3. **Take the highest power of each prime factor**:
- For \(2\): The highest power is \(2^1\) (from the first number).
- For \(3\): The highest power is \(3^1\) (present in both numbers).
- For \(5\): The highest power is \(5^1\) (present in both numbers).
- For \(7\): The highest power is \(7^1\) (present in both numbers).
- For \(11\): The highest power is \(11^1\) (from the second number).

4. **Multiply these together to find the LCM**:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^1 \times 7^1 \times 11^1 \]
\[ \text{LCM} = 2 \times 3 \times 5 \times 7 \times 11 \]

5. **Calculate the LCM**:
- First, calculate \(2 \times 3 = 6\).
- Then, \(6 \times 5 = 30\).
- Next, \(30 \times 7 = 210\).
- Finally, \(210 \times 11 = 2310\).

Thus, the LCM of \(2 \times 3 \times 5 \times 7\) and \(3 \times 5 \times 7 \times 11\) is \(2310\).

### Final Answer:
\[ \text{LCM} = 2310 \]