Find the LCM of `(2^5 xx 3 xx 5^2 xx 7) (2^4 xx 3^3 xx 5 xx 7^2 xx 11) and (2 xx 3^5xx 5^4)`

Options

A. `2^4 xx 3^3 xx 5^4`
B. `2xx3xx7xx5xx11`
C. `2^4 xx 3^3 xx 5^4 xx 7^2 xx 11`
D. `2^4 xx 3^4 xx 5^4 xx 7`

Answer

To find the LCM (Least Common Multiple) of the given numbers, we will follow these steps:

### Step 1: Identify the prime factorization of each number.

1. **First number (A)**:
\[ A = 2^5 \times 3^1 \times 5^2 \times 7^1 \]

2. **Second number (B)**:
\[ B = 2^4 \times 3^3 \times 5^1 \times 7^2 \times 11^1 \]

3. **Third number (C)**:
\[ C = 2^1 \times 3^5 \times 5^4 \]

### Step 2: Determine the highest power of each prime factor present in the factorizations.

- For \(2\):
- Highest power from A: \(5\)
- Highest power from B: \(4\)
- Highest power from C: \(1\)
- **Maximum**: \(2^5\)

- For \(3\):
- Highest power from A: \(1\)
- Highest power from B: \(3\)
- Highest power from C: \(5\)
- **Maximum**: \(3^5\)

- For \(5\):
- Highest power from A: \(2\)
- Highest power from B: \(1\)
- Highest power from C: \(4\)
- **Maximum**: \(5^4\)

- For \(7\):
- Highest power from A: \(1\)
- Highest power from B: \(2\)
- Highest power from C: \(0\) (since \(C\) does not have \(7\))
- **Maximum**: \(7^2\)

- For \(11\):
- Highest power from A: \(0\) (since \(A\) does not have \(11\))
- Highest power from B: \(1\)
- Highest power from C: \(0\) (since \(C\) does not have \(11\))
- **Maximum**: \(11^1\)

### Step 3: Write the LCM using the highest powers found.

The LCM is given by:
\[ \text{LCM} = 2^5 \times 3^5 \times 5^4 \times 7^2 \times 11^1 \]

### Step 4: Final answer.

Thus, the LCM of the given numbers is:
\[ \text{LCM} = 2^5 \times 3^5 \times 5^4 \times 7^2 \times 11 \]

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