The product of LCM and HCF of `(3)/(5) and (8)/(3)` is

Options

A. `(8)/(5)`
B. `(5)/(8)`
C. `(1)/(8)`
D. `(5)/(3)`

Answer

To find the product of the LCM and HCF of the fractions \( \frac{3}{5} \) and \( \frac{8}{3} \), we will follow these steps:

### Step 1: Find the LCM of the numerators
The numerators of the fractions are 3 and 8. To find the LCM of 3 and 8:
- The prime factorization of 3 is \( 3^1 \).
- The prime factorization of 8 is \( 2^3 \).

The LCM is found by taking the highest power of each prime factor:
- For 2, the highest power is \( 2^3 \).
- For 3, the highest power is \( 3^1 \).

Thus,
\[ \text{LCM}(3, 8) = 2^3 \times 3^1 = 8 \times 3 = 24. \]

### Step 2: Find the HCF of the denominators
The denominators of the fractions are 5 and 3. To find the HCF of 5 and 3:
- The prime factorization of 5 is \( 5^1 \).
- The prime factorization of 3 is \( 3^1 \).

Since there are no common prime factors, the HCF is:
\[ \text{HCF}(5, 3) = 1. \]

### Step 3: Calculate LCM of the fractions
Now, we can find the LCM of the fractions:
\[ \text{LCM}\left(\frac{3}{5}, \frac{8}{3}\right) = \frac{\text{LCM}(3, 8)}{\text{HCF}(5, 3)} = \frac{24}{1} = 24. \]

### Step 4: Find the HCF of the fractions
Next, we find the HCF of the fractions:
\[ \text{HCF}\left(\frac{3}{5}, \frac{8}{3}\right) = \frac{\text{HCF}(3, 8)}{\text{LCM}(5, 3)}. \]
- The HCF of 3 and 8 is \( 1 \) (since they have no common factors).
- The LCM of 5 and 3 is \( 15 \) (since \( 5 \times 3 = 15 \)).

Thus,
\[ \text{HCF}\left(\frac{3}{5}, \frac{8}{3}\right) = \frac{1}{15}. \]

### Step 5: Calculate the product of LCM and HCF
Now we can find the product of the LCM and HCF:
\[ \text{Product} = \text{LCM} \times \text{HCF} = 24 \times \frac{1}{15} = \frac{24}{15}. \]

### Step 6: Simplify the fraction
To simplify \( \frac{24}{15} \):
- Divide both the numerator and denominator by their HCF, which is 3:
\[ \frac{24 \div 3}{15 \div 3} = \frac{8}{5}. \]

### Final Answer
Thus, the product of the LCM and HCF of \( \frac{3}{5} \) and \( \frac{8}{3} \) is \( \frac{8}{5} \).

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