The LCM of `2/3,4/9,5/6,7/12` is

Options

A. `1/18`
B. `1/36`
C. `35/9`
D. `140/3`

Answer

To find the LCM of the fractions \( \frac{2}{3}, \frac{4}{9}, \frac{5}{6}, \frac{7}{12} \), we will follow these steps:

### Step 1: Identify the numerators and denominators
The numerators are:
- 2
- 4
- 5
- 7

The denominators are:
- 3
- 9
- 6
- 12

### Step 2: Find the LCM of the numerators
To find the LCM of the numerators (2, 4, 5, 7), we will factor each number:
- 2 = \( 2^1 \)
- 4 = \( 2^2 \)
- 5 = \( 5^1 \)
- 7 = \( 7^1 \)

The LCM is found by taking the highest power of each prime factor:
- For 2, the highest power is \( 2^2 \)
- For 5, the highest power is \( 5^1 \)
- For 7, the highest power is \( 7^1 \)

Thus,
\[ \text{LCM} = 2^2 \times 5^1 \times 7^1 = 4 \times 5 \times 7 = 140 \]

### Step 3: Find the SCF (GCD) of the denominators
To find the SCF of the denominators (3, 9, 6, 12), we will factor each number:
- 3 = \( 3^1 \)
- 9 = \( 3^2 \)
- 6 = \( 2^1 \times 3^1 \)
- 12 = \( 2^2 \times 3^1 \)

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

Thus,
\[ \text{SCF} = 3^1 = 3 \]

### Step 4: Calculate the LCM of the fractions
The LCM of the fractions is given by the formula:
\[ \text{LCM} = \frac{\text{LCM of numerators}}{\text{SCF of denominators}} = \frac{140}{3} \]

### Final Answer
The LCM of \( \frac{2}{3}, \frac{4}{9}, \frac{5}{6}, \frac{7}{12} \) is \( \frac{140}{3} \).

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