Least common multiple of $\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{9}{13}$ is :

Question

Accepted Answer

To determine the least common multiple (LCM) of the fractions $\frac{2}{3}$, $\frac{3}{5}$, $\frac{4}{7}$, and $\frac{9}{13}$, we need to consider both the numerators and the denominators separately.

### Step-by-Step Explanation:

1. **Understanding LCM of Fractions:**
   The LCM of fractions is found by taking the LCM of the numerators and the greatest common divisor (GCD) of the denominators. The formula is:
   $$
   \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{GCD of denominators}}
   $$

2. **Numerators:**
   The numerators are 2, 3, 4, and 9. We need to find the LCM of these numbers.
   - Prime factorization:
     - 2 is $2$
     - 3 is $3$
     - 4 is $2^2$
     - 9 is $3^2$
   - LCM is found by taking the highest power of each prime:
     - Highest power of 2 is $2^2$
     - Highest power of 3 is $3^2$
   - Therefore, LCM of numerators = $2^2 \times 3^2 = 4 \times 9 = 36$

3. **Denominators:**
   The denominators are 3, 5, 7, and 13. We need to find the GCD of these numbers.
   - Since 3, 5, 7, and 13 are all prime numbers and do not share any common factors other than 1, the GCD is 1.

4. **Calculating LCM of Fractions:**
   Using the formula:
   $$
   \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{GCD of denominators}} = \frac{36}{1} = 36
   $$

### Conclusion:

- **Option 4: 36** is indeed the correct answer as it matches the calculated LCM of the given fractions.

### Explanation of Incorrect Options:

1. **Option 1: $\frac{20}{3}$**
   - This option does not match the calculated LCM of 36. The LCM of the numerators is 36, and the GCD of the denominators is 1, leading to a result of 36, not $\frac{20}{3}$.

2. **Option 2: $\frac{18}{7}$**
   - Similar to Option 1, this fraction does not align with the calculated LCM. The LCM of the numerators is 36, and the GCD of the denominators is 1, resulting in 36, not $\frac{18}{7}$.

3. **Option 3: 42**
   - While 42 is a whole number, it does not match the calculated LCM of 36. The correct LCM of the numerators is 36, and the GCD of the denominators is 1, leading to a final result of 36.

By following the logical steps and calculations, we confirm that Option 4: 36 is the correct answer for the least common multiple of the given fractions.