Find the LCM of the following fractions , 2/3 , 8/9 , 16/27 , 32/81

Options

A. 32/81
B. 81/32
C. 32/3
D. `11//41`

Answer

To find the LCM of the fractions \( \frac{2}{3}, \frac{8}{9}, \frac{16}{27}, \frac{32}{81} \), we can follow these steps:

### Step 1: Identify the numerators and denominators
The fractions are:
- Numerators: 2, 8, 16, 32
- Denominators: 3, 9, 27, 81

### Step 2: Find the LCM of the numerators
We need to find the LCM of the numerators: 2, 8, 16, and 32.

- The prime factorization of each number is:
- \( 2 = 2^1 \)
- \( 8 = 2^3 \)
- \( 16 = 2^4 \)
- \( 32 = 2^5 \)

The LCM is found by taking the highest power of each prime factor:
- The highest power of 2 is \( 2^5 \).

Thus, the LCM of the numerators is:
\[ \text{LCM} = 32 \]

### Step 3: Find the SCF (GCD) of the denominators
Now we need to find the SCF (or GCD) of the denominators: 3, 9, 27, and 81.

- The prime factorization of each number is:
- \( 3 = 3^1 \)
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)
- \( 81 = 3^4 \)

The SCF is found by taking the lowest power of each prime factor:
- The lowest power of 3 is \( 3^1 \).

Thus, the SCF of the denominators is:
\[ \text{SCF} = 3 \]

### Step 4: Write the final answer
Now, we can write the LCM of the fractions:
\[ \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{SCF of denominators}} = \frac{32}{3} \]

### Final Answer:
\[ \frac{32}{3} \]