LCM of `3/5,6/25,9/5,12/15`

Options

A. `12/5`
B. `12/25`
C. `36/25`
D. `36/5`

Answer

To find the LCM of the rational numbers \( \frac{3}{5}, \frac{6}{25}, \frac{9}{5}, \frac{12}{15} \), we will follow the rule that states:

\[ \text{LCM of a rational number} = \frac{\text{LCM of the numerators}}{\text{HCF of the denominators}} \]

### Step 1: Identify the numerators and denominators
The numerators are \( 3, 6, 9, 12 \) and the denominators are \( 5, 25, 5, 15 \).

### Step 2: Find the LCM of the numerators
To find the LCM of \( 3, 6, 9, 12 \):
- The prime factorization is:
- \( 3 = 3^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 9 = 3^2 \)
- \( 12 = 2^2 \times 3^1 \)

- The LCM is found by taking the highest power of each prime:
- For \( 2 \): highest power is \( 2^2 \) from \( 12 \)
- For \( 3 \): highest power is \( 3^2 \) from \( 9 \)

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 3: Find the HCF of the denominators
To find the HCF of \( 5, 25, 5, 15 \):
- The prime factorization is:
- \( 5 = 5^1 \)
- \( 25 = 5^2 \)
- \( 15 = 3^1 \times 5^1 \)

- The HCF is found by taking the lowest power of each prime:
- For \( 5 \): lowest power is \( 5^1 \)
- For \( 3 \): it does not appear in all, so we ignore it.

Thus, the HCF is:
\[ \text{HCF} = 5^1 = 5 \]

### Step 4: Calculate the LCM of the rational numbers
Now, we can find the LCM of the rational numbers:
\[ \text{LCM} = \frac{\text{LCM of the numerators}}{\text{HCF of the denominators}} = \frac{36}{5} \]

### Final Answer
The LCM of \( \frac{3}{5}, \frac{6}{25}, \frac{9}{5}, \frac{12}{15} \) is:
\[ \frac{36}{5} \]

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