Calculate the LCM of 72/250,126/75 and 162/165

Answer

To calculate the LCM of the fractions \( \frac{72}{250}, \frac{126}{75}, \) and \( \frac{162}{165} \), we can follow these steps:

### Step 1: Simplify Each Fraction
First, we simplify each fraction to its lowest terms.

1. **For \( \frac{72}{250} \)**:
- The GCD of 72 and 250 is 2.
- So, \( \frac{72 \div 2}{250 \div 2} = \frac{36}{125} \).

2. **For \( \frac{126}{75} \)**:
- The GCD of 126 and 75 is 3.
- So, \( \frac{126 \div 3}{75 \div 3} = \frac{42}{25} \).

3. **For \( \frac{162}{165} \)**:
- The GCD of 162 and 165 is 3.
- So, \( \frac{162 \div 3}{165 \div 3} = \frac{54}{55} \).

Now we have the simplified fractions:
- \( \frac{36}{125}, \frac{42}{25}, \frac{54}{55} \)

### Step 2: Find the LCM of the Numerators
Next, we find the LCM of the numerators: 36, 42, and 54.

1. **Prime Factorization**:
- \( 36 = 2^2 \times 3^2 \)
- \( 42 = 2^1 \times 3^1 \times 7^1 \)
- \( 54 = 2^1 \times 3^3 \)

2. **LCM Calculation**:
- Take the highest power of each prime:
- For 2: \( 2^2 \)
- For 3: \( 3^3 \)
- For 7: \( 7^1 \)

So, the LCM of the numerators is:
\[ LCM = 2^2 \times 3^3 \times 7^1 = 4 \times 27 \times 7 = 756 \]

### Step 3: Find the GCD of the Denominators
Now, we find the GCD of the denominators: 125, 25, and 55.

1. **Prime Factorization**:
- \( 125 = 5^3 \)
- \( 25 = 5^2 \)
- \( 55 = 5^1 \times 11^1 \)

2. **GCD Calculation**:
- The only common prime factor is 5, and the lowest power is \( 5^1 \).
So, the GCD of the denominators is:
\[ GCD = 5^1 = 5 \]

### Step 4: Calculate the LCM of the Fractions
Now we can use the formula for the LCM of fractions:
\[ LCM\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{LCM(a, c, e)}{GCD(b, d, f)} \]
Substituting the values we found:
\[ LCM\left(\frac{36}{125}, \frac{42}{25}, \frac{54}{55}\right) = \frac{756}{5} \]

### Step 5: Final Result
Thus, the LCM of the fractions \( \frac{72}{250}, \frac{126}{75}, \) and \( \frac{162}{165} \) is:
\[ \frac{756}{5} \]