Find the LCM of `2/3 , 3/5 , 4/7 and 9/(13)`

Options

A. 36
B. `1/(36)`
C. `1/(1365)`
D. `(12)/(455)`

Answer

To find the LCM of the fractions \( \frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{9}{13} \), we will follow these steps:

### Step 1: Identify the numerators and denominators
The fractions are:
- Numerators: \( 2, 3, 4, 9 \)
- Denominators: \( 3, 5, 7, 13 \)

### Step 2: Find the LCM of the numerators
To find the LCM of the numerators \( 2, 3, 4, 9 \):
- The prime factorization is:
- \( 2 = 2^1 \)
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 9 = 3^2 \)

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

Thus, the LCM of the numerators is:
\[ LCM(2, 3, 4, 9) = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 3: Find the LCM of the denominators
To find the LCM of the denominators \( 3, 5, 7, 13 \):
- All these numbers are prime.

Thus, the LCM is simply the product of these primes:
\[ LCM(3, 5, 7, 13) = 3 \times 5 \times 7 \times 13 \]

Calculating this step-by-step:
- \( 3 \times 5 = 15 \)
- \( 15 \times 7 = 105 \)
- \( 105 \times 13 = 1365 \)

So, the LCM of the denominators is:
\[ LCM(3, 5, 7, 13) = 1365 \]

### Step 4: Form the LCM of the fractions
The LCM of the fractions is given by:
\[ LCM\left(\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{9}{13}\right) = \frac{LCM(\text{numerators})}{LCM(\text{denominators})} = \frac{36}{1365} \]

### Final Answer
The LCM of \( \frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{9}{13} \) is:
\[ \frac{36}{1365} \]

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