What is the L.C.M of `(10/21,20/63,55/56)` ?

Options

A. `{:(5)/(7):}`
B. `{:(220)/(7):}`
C. `{:(220)/(504):}`
D. `{:(5)/(504):}`

Answer

To find the L.C.M of the fractions \( \frac{10}{21}, \frac{20}{63}, \frac{55}{56} \), we follow these steps:

### Step 1: Identify the Numerators and Denominators
The fractions given are:
- Numerators: 10, 20, 55
- Denominators: 21, 63, 56

### Step 2: Calculate the L.C.M of the Numerators
We need to find the L.C.M of the numerators: 10, 20, and 55.

**Finding L.C.M:**
- The prime factorization of 10 is \( 2 \times 5 \).
- The prime factorization of 20 is \( 2^2 \times 5 \).
- The prime factorization of 55 is \( 5 \times 11 \).

Now, we take the highest power of each prime:
- For 2: \( 2^2 \) (from 20)
- For 5: \( 5^1 \) (common in all)
- For 11: \( 11^1 \) (from 55)

Thus, the L.C.M is:
\[ L.C.M = 2^2 \times 5^1 \times 11^1 = 4 \times 5 \times 11 = 220 \]

### Step 3: Calculate the H.C.F of the Denominators
Next, we find the H.C.F of the denominators: 21, 63, and 56.

**Finding H.C.F:**
- The prime factorization of 21 is \( 3 \times 7 \).
- The prime factorization of 63 is \( 3^2 \times 7 \).
- The prime factorization of 56 is \( 2^3 \times 7 \).

Now, we take the lowest power of the common prime:
- The only common prime factor is 7.

Thus, the H.C.F is:
\[ H.C.F = 7 \]

### Step 4: Form the Final L.C.M of the Fractions
The L.C.M of the fractions is given by the formula:
\[ \text{L.C.M} \left( \frac{a}{b}, \frac{c}{d}, \frac{e}{f} \right) = \frac{\text{L.C.M of numerators}}{\text{H.C.F of denominators}} \]
Substituting the values we calculated:
\[ \text{L.C.M} = \frac{220}{7} \]

### Final Answer
The L.C.M of \( \frac{10}{21}, \frac{20}{63}, \frac{55}{56} \) is \( \frac{220}{7} \).

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