The LCM of `59/749, 651/60 and 448/895` is

Options

A. `2458176`
B. `2458247`
C. `5478176`
D. `5874176`

Answer

To find the LCM of the rational numbers \( \frac{59}{749}, \frac{651}{60}, \) and \( \frac{448}{895} \), we can follow these steps:

### Step 1: Identify the Numerators and Denominators
The rational numbers are:
- Numerators: \( 59, 651, 448 \)
- Denominators: \( 749, 60, 895 \)

### Step 2: Find the LCM of the Numerators
To find the LCM of the numerators \( 59, 651, \) and \( 448 \), we first factor each number:
- \( 59 \) is a prime number.
- \( 651 = 3 \times 7 \times 31 \)
- \( 448 = 2^6 \times 7 \)

Now, we take the highest power of each prime factor:
- From \( 59 \): \( 59^1 \)
- From \( 651 \): \( 3^1, 7^1, 31^1 \)
- From \( 448 \): \( 2^6 \)

Thus, the LCM of the numerators is:
\[ LCM = 2^6 \times 3^1 \times 7^1 \times 31^1 \times 59^1 \]

### Step 3: Find the HCF of the Denominators
Next, we find the HCF of the denominators \( 749, 60, \) and \( 895 \):
- \( 749 = 7 \times 107 \)
- \( 60 = 2^2 \times 3 \times 5 \)
- \( 895 = 5 \times 179 \)

The only common factor among these numbers is \( 1 \). Therefore, the HCF of the denominators is:
\[ HCF = 1 \]

### Step 4: Calculate the LCM of the Rational Numbers
The LCM of the rational numbers is given by the formula:
\[ LCM\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{LCM(a, c, e)}{HCF(b, d, f)} \]
Substituting the values we found:
\[ LCM = \frac{2^6 \times 3^1 \times 7^1 \times 31^1 \times 59^1}{1} \]

### Step 5: Calculate the Final Value
Now we need to calculate the value of \( LCM \):
\[ LCM = 64 \times 3 \times 7 \times 31 \times 59 \]
Calculating step-by-step:
1. \( 64 \times 3 = 192 \)
2. \( 192 \times 7 = 1344 \)
3. \( 1344 \times 31 = 41664 \)
4. \( 41664 \times 59 = 2458176 \)

Thus, the LCM of the given rational numbers is:
\[ \boxed{2458176} \]