The LCM of `(2^(4)xx3^(6)xx5^(3)),(3^(8)xx5^(9)xx7^(5)),(2^(10)xx3^(5)xx11^(4))`

Options

A. A)`(3^(6)xx2^(4)xx5^(3)xx7^(5))`
B. B)`(2^(3)xx3^(4)xx5^(4))`
C. C)`(2^(10)xx3^(8)xx5^(9)xx7^(5)xx11^(4))`
D. D)`(2^(8)xx3^(9)xx7^(5)xx11^(4))`

Answer

To find the LCM (Least Common Multiple) of the numbers \( (2^4 \times 3^6 \times 5^3), (3^8 \times 5^9 \times 7^5), (2^{10} \times 3^5 \times 11^4) \), we will follow these steps:

### Step 1: Identify the prime factors
The prime factors present in the numbers are:
- 2
- 3
- 5
- 7
- 11

### Step 2: Determine the highest power of each prime factor
Now, we will find the highest power of each prime factor from the given numbers:

1. **For 2**:
- In \( 2^4 \), the power is 4.
- In \( 3^8 \times 5^9 \times 7^5 \), there is no 2, so the power is 0.
- In \( 2^{10} \), the power is 10.
- Highest power of 2 = \( \max(4, 0, 10) = 10 \).

2. **For 3**:
- In \( 2^4 \times 3^6 \times 5^3 \), the power is 6.
- In \( 3^8 \times 5^9 \times 7^5 \), the power is 8.
- In \( 2^{10} \times 3^5 \times 11^4 \), the power is 5.
- Highest power of 3 = \( \max(6, 8, 5) = 8 \).

3. **For 5**:
- In \( 2^4 \times 3^6 \times 5^3 \), the power is 3.
- In \( 3^8 \times 5^9 \times 7^5 \), the power is 9.
- In \( 2^{10} \times 3^5 \times 11^4 \), the power is 0.
- Highest power of 5 = \( \max(3, 9, 0) = 9 \).

4. **For 7**:
- In \( 2^4 \times 3^6 \times 5^3 \), there is no 7, so the power is 0.
- In \( 3^8 \times 5^9 \times 7^5 \), the power is 5.
- In \( 2^{10} \times 3^5 \times 11^4 \), there is no 7, so the power is 0.
- Highest power of 7 = \( \max(0, 5, 0) = 5 \).

5. **For 11**:
- In \( 2^4 \times 3^6 \times 5^3 \), there is no 11, so the power is 0.
- In \( 3^8 \times 5^9 \times 7^5 \), there is no 11, so the power is 0.
- In \( 2^{10} \times 3^5 \times 11^4 \), the power is 4.
- Highest power of 11 = \( \max(0, 0, 4) = 4 \).

### Step 3: Write the LCM
Now that we have the highest powers, we can write the LCM as:
\[ \text{LCM} = 2^{10} \times 3^8 \times 5^9 \times 7^5 \times 11^4 \]

### Final Answer
Thus, the LCM of the given numbers is:
\[ \text{LCM} = 2^{10} \times 3^8 \times 5^9 \times 7^5 \times 11^4 \]

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