L.C.M of `2/7, 3/(14)` and `5/3` is -

Options

A. 45
B. 35
C. 30
D. 25

Answer

To find the L.C.M. of the fractions \( \frac{2}{7}, \frac{3}{14}, \) and \( \frac{5}{3} \), we will follow these steps:

### Step 1: Identify the Numerators and Denominators
The numerators of the fractions are:
- \( 2 \) (from \( \frac{2}{7} \))
- \( 3 \) (from \( \frac{3}{14} \))
- \( 5 \) (from \( \frac{5}{3} \))

The denominators of the fractions are:
- \( 7 \) (from \( \frac{2}{7} \))
- \( 14 \) (from \( \frac{3}{14} \))
- \( 3 \) (from \( \frac{5}{3} \))

### Step 2: Calculate the L.C.M. of the Numerators
To find the L.C.M. of the numerators \( 2, 3, \) and \( 5 \):
- Since all these numbers are prime, the L.C.M. is simply their product:
\[ \text{L.C.M. of numerators} = 2 \times 3 \times 5 = 30 \]

### Step 3: Calculate the L.C.M. of the Denominators
To find the L.C.M. of the denominators \( 7, 14, \) and \( 3 \):
- The prime factorization of the denominators is:
- \( 7 = 7^1 \)
- \( 14 = 2^1 \times 7^1 \)
- \( 3 = 3^1 \)

To find the L.C.M., we take the highest power of each prime:
- For \( 2 \): \( 2^1 \) (from \( 14 \))
- For \( 3 \): \( 3^1 \) (from \( 3 \))
- For \( 7 \): \( 7^1 \) (from both \( 7 \) and \( 14 \))

Thus, the L.C.M. of the denominators is:
\[ \text{L.C.M. of denominators} = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42 \]

### Step 4: Calculate the L.C.M. of the Fractions
The formula for the L.C.M. of fractions is:
\[ \text{L.C.M. of fractions} = \frac{\text{L.C.M. of numerators}}{\text{H.C.F. of denominators}} \]
Since the denominators are \( 7, 14, \) and \( 3 \), we need to find the H.C.F. of the denominators:
- The H.C.F. of \( 7, 14, \) and \( 3 \) is \( 1 \) (since there are no common factors).

Now we can calculate the L.C.M. of the fractions:
\[ \text{L.C.M. of fractions} = \frac{30}{1} = 30 \]

### Final Answer
The L.C.M. of \( \frac{2}{7}, \frac{3}{14}, \) and \( \frac{5}{3} \) is \( 30 \).

---