Find the smallest positive number which is exactly divisible by `1/3, 1/2, 3/7 and 4/11`.

Answer

To find the smallest positive number that is exactly divisible by \( \frac{1}{3}, \frac{1}{2}, \frac{3}{7}, \) and \( \frac{4}{11} \), we can follow these steps:

### Step 1: Identify the Numerators and Denominators
We need to extract the numerators and denominators from the given fractions:
- Numerators: \( 1, 1, 3, 4 \)
- Denominators: \( 3, 2, 7, 11 \)

### Step 2: Calculate the LCM of the Numerators
We will find the Least Common Multiple (LCM) of the numerators \( 1, 1, 3, 4 \).

- The prime factorization of the numbers:
- \( 1 = 1 \)
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)

To find the LCM, we take the highest power of each prime:
- For \( 2 \): highest power is \( 2^2 \)
- For \( 3 \): highest power is \( 3^1 \)

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Step 3: Calculate the HCF of the Denominators
Next, we will find the Highest Common Factor (HCF) of the denominators \( 3, 2, 7, 11 \).

- Since all these numbers are prime and do not share any common factors other than \( 1 \), the HCF is:
\[ \text{HCF} = 1 \]

### Step 4: Use the Formula to Find the Smallest Positive Number
Now, we can use the formula:
\[ \text{Smallest Positive Number} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}} \]
Substituting the values we found:
\[ \text{Smallest Positive Number} = \frac{12}{1} = 12 \]

### Final Answer
The smallest positive number which is exactly divisible by \( \frac{1}{3}, \frac{1}{2}, \frac{3}{7}, \) and \( \frac{4}{11} \) is **12**.

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