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

Options

A. `2 xx 23 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 expressions \(2 \times 3 \times 5 \times 7\) and \(3 \times 5 \times 7 \times 11\), we will follow these steps:

### Step 1: Identify the prime factors
The prime factorization of the first number is:
- \(2 \times 3 \times 5 \times 7\)

The prime factorization of the second number is:
- \(3 \times 5 \times 7 \times 11\)

### Step 2: List all unique prime factors
From both numbers, we identify the unique prime factors:
- From the first number: \(2, 3, 5, 7\)
- From the second number: \(3, 5, 7, 11\)

Combining these, the unique prime factors are:
- \(2, 3, 5, 7, 11\)

### Step 3: Determine the highest power of each prime factor
Next, we take the highest power of each prime factor from both numbers:
- For \(2\): The highest power is \(2^1\) (from the first number).
- For \(3\): The highest power is \(3^1\) (appears in both numbers).
- For \(5\): The highest power is \(5^1\) (appears in both numbers).
- For \(7\): The highest power is \(7^1\) (appears in both numbers).
- For \(11\): The highest power is \(11^1\) (from the second number).

### Step 4: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^1 \times 7^1 \times 11^1 \]
Calculating this gives:
\[ \text{LCM} = 2 \times 3 \times 5 \times 7 \times 11 \]

### Step 5: Perform the multiplication
Calculating step by step:
1. \(2 \times 3 = 6\)
2. \(6 \times 5 = 30\)
3. \(30 \times 7 = 210\)
4. \(210 \times 11 = 2310\)

Thus, the LCM of \(2 \times 3 \times 5 \times 7\) and \(3 \times 5 \times 7 \times 11\) is:
\[ \text{LCM} = 2310 \]

### Final Answer
The LCM of \(2 \times 3 \times 5 \times 7\) and \(3 \times 5 \times 7 \times 11\) is **2310**.

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