The LCM of `(1)/(3), (2)/(9), (5)/(6) and (4)/(27)` is equal to

Options

A. `(1)/(54)`
B. `(10)/(27)`
C. `(20)/(3)`
D. `(23)/(4)`

Answer

To find the LCM of the fractions \( \frac{1}{3}, \frac{2}{9}, \frac{5}{6}, \frac{4}{27} \), we will follow these steps:

### Step 1: Identify the Numerators and Denominators
The fractions are:
- Numerators: \( 1, 2, 5, 4 \)
- Denominators: \( 3, 9, 6, 27 \)

### Step 2: Calculate the LCM of the Numerators
To find the LCM of the numerators \( 1, 2, 5, 4 \):
- The prime factorization is:
- \( 1 = 1 \)
- \( 2 = 2 \)
- \( 5 = 5 \)
- \( 4 = 2^2 \)

The LCM is found by taking the highest power of each prime:
- The highest power of \( 2 \) is \( 2^2 \)
- The highest power of \( 5 \) is \( 5^1 \)

Thus, the LCM of the numerators is:
\[ LCM = 2^2 \times 5 = 4 \times 5 = 20 \]

### Step 3: Calculate the HCF of the Denominators
To find the HCF of the denominators \( 3, 9, 6, 27 \):
- The prime factorization is:
- \( 3 = 3^1 \)
- \( 9 = 3^2 \)
- \( 6 = 3^1 \times 2^1 \)
- \( 27 = 3^3 \)

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

Thus, the HCF of the denominators is:
\[ HCF = 3^1 = 3 \]

### Step 4: Calculate the LCM of the Fractions
The LCM of the fractions is given by the formula:
\[ LCM\left(\frac{a}{b}\right) = \frac{LCM(a)}{HCF(b)} \]
Using our results:
\[ LCM\left(\frac{1}{3}, \frac{2}{9}, \frac{5}{6}, \frac{4}{27}\right) = \frac{LCM(1, 2, 5, 4)}{HCF(3, 9, 6, 27)} = \frac{20}{3} \]

### Final Answer
The LCM of \( \frac{1}{3}, \frac{2}{9}, \frac{5}{6}, \frac{4}{27} \) is \( \frac{20}{3} \).

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