To find the LCM of the fractions \( \frac{2}{3}, \frac{7}{9}, \frac{14}{15} \), we can follow these steps:
### Step 1: Identify the numerators and denominators
The numerators are \( 2, 7, \) and \( 14 \).
The denominators are \( 3, 9, \) and \( 15 \).
### Step 2: Calculate the LCM of the numerators
To find the LCM of \( 2, 7, \) and \( 14 \):
- The prime factorization of \( 2 \) is \( 2^1 \).
- The prime factorization of \( 7 \) is \( 7^1 \).
- The prime factorization of \( 14 \) is \( 2^1 \times 7^1 \).
The LCM is found by taking the highest power of each prime:
- For \( 2 \): highest power is \( 2^1 \).
- For \( 7 \): highest power is \( 7^1 \).
Thus, the LCM of \( 2, 7, \) and \( 14 \) is:
\[
LCM = 2^1 \times 7^1 = 14
\]
### Step 3: Calculate the GCD of the denominators
To find the GCD of \( 3, 9, \) and \( 15 \):
- The prime factorization of \( 3 \) is \( 3^1 \).
- The prime factorization of \( 9 \) is \( 3^2 \).
- The prime factorization of \( 15 \) is \( 3^1 \times 5^1 \).
The GCD is found by taking the lowest power of each prime:
- For \( 3 \): lowest power is \( 3^1 \).
Thus, the GCD of \( 3, 9, \) and \( 15 \) is:
\[
GCD = 3^1 = 3
\]
### Step 4: Calculate the LCM of the fractions
The LCM of the fractions is given by the formula:
\[
LCM\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{LCM(a, c, e)}{GCD(b, d, f)}
\]
Substituting the values we found:
\[
LCM\left(\frac{2}{3}, \frac{7}{9}, \frac{14}{15}\right) = \frac{14}{3}
\]
### Final Answer
The LCM of \( \frac{2}{3}, \frac{7}{9}, \frac{14}{15} \) is \( \frac{14}{3} \).
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