What will be the Least Common Multiple of `2^(2) xx 3^(5) xx 7^(3) and 2^(4) xx 3^(4) xx 7^(1)`?

Options

A. `2^(2) xx 3^(4) xx 7^(1)`
B. `2^(6) xx 3^(9) xx 7^(4)`
C. `2^(4) xx 3^(6) xx 7^(4)`
D. `2^(4) xx 3^(5) xx 7^(3)`

Answer

To find the Least Common Multiple (LCM) of the two expressions \(2^2 \times 3^5 \times 7^3\) and \(2^4 \times 3^4 \times 7^1\), we follow these steps:

### Step 1: Identify the prime factors and their powers in both expressions.
- For the first expression \(2^2 \times 3^5 \times 7^3\):
- The prime factor 2 has a power of 2.
- The prime factor 3 has a power of 5.
- The prime factor 7 has a power of 3.

- For the second expression \(2^4 \times 3^4 \times 7^1\):
- The prime factor 2 has a power of 4.
- The prime factor 3 has a power of 4.
- The prime factor 7 has a power of 1.

### Step 2: Determine the maximum power for each prime factor.
- For the prime factor 2:
- Maximum power = max(2, 4) = 4
- For the prime factor 3:
- Maximum power = max(5, 4) = 5
- For the prime factor 7:
- Maximum power = max(3, 1) = 3

### Step 3: Write the LCM using the maximum powers of the prime factors.
The LCM is given by:
\[ LCM = 2^{\text{max power of 2}} \times 3^{\text{max power of 3}} \times 7^{\text{max power of 7}} \]
Substituting the maximum powers we found:
\[ LCM = 2^4 \times 3^5 \times 7^3 \]

### Step 4: Finalize the LCM expression.
Thus, the Least Common Multiple of \(2^2 \times 3^5 \times 7^3\) and \(2^4 \times 3^4 \times 7^1\) is:
\[ LCM = 2^4 \times 3^5 \times 7^3 \]