Find the LCM of 22, 54, 108, 135 and 198.

Answer

To find the LCM (Least Common Multiple) of the numbers 22, 54, 108, 135, and 198 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of Each Number
- **22**:
- Divide by 2: \( 22 \div 2 = 11 \)
- Prime factors: \( 2 \times 11 \)

- **54**:
- Divide by 2: \( 54 \div 2 = 27 \)
- Divide by 3: \( 27 \div 3 = 9 \)
- Divide by 3: \( 9 \div 3 = 3 \)
- Divide by 3: \( 3 \div 3 = 1 \)
- Prime factors: \( 2 \times 3^3 \)

- **108**:
- Divide by 2: \( 108 \div 2 = 54 \)
- Divide by 2: \( 54 \div 2 = 27 \)
- Divide by 3: \( 27 \div 3 = 9 \)
- Divide by 3: \( 9 \div 3 = 3 \)
- Divide by 3: \( 3 \div 3 = 1 \)
- Prime factors: \( 2^2 \times 3^3 \)

- **135**:
- Divide by 3: \( 135 \div 3 = 45 \)
- Divide by 3: \( 45 \div 3 = 15 \)
- Divide by 3: \( 15 \div 3 = 5 \)
- Divide by 5: \( 5 \div 5 = 1 \)
- Prime factors: \( 3^3 \times 5 \)

- **198**:
- Divide by 2: \( 198 \div 2 = 99 \)
- Divide by 3: \( 99 \div 3 = 33 \)
- Divide by 3: \( 33 \div 3 = 11 \)
- Divide by 11: \( 11 \div 11 = 1 \)
- Prime factors: \( 2 \times 3^2 \times 11 \)

### Step 2: List the Highest Powers of Each Prime Factor
Now, we will take the highest power of each prime factor from all the numbers:
- From 22: \( 2^1 \), \( 11^1 \)
- From 54: \( 2^1 \), \( 3^3 \)
- From 108: \( 2^2 \), \( 3^3 \)
- From 135: \( 3^3 \), \( 5^1 \)
- From 198: \( 2^1 \), \( 3^2 \), \( 11^1 \)

The highest powers are:
- \( 2^2 \) (from 108)
- \( 3^3 \) (from 54, 108, and 135)
- \( 5^1 \) (from 135)
- \( 11^1 \) (from 22 and 198)

### Step 3: Calculate the LCM
Now, multiply these highest powers together to find the LCM:
\[ LCM = 2^2 \times 3^3 \times 5^1 \times 11^1 \]
Calculating this step-by-step:
- \( 2^2 = 4 \)
- \( 3^3 = 27 \)
- \( 5^1 = 5 \)
- \( 11^1 = 11 \)

Now, multiply these results:
\[ 4 \times 27 = 108 \]
\[ 108 \times 5 = 540 \]
\[ 540 \times 11 = 5940 \]

### Final Answer
Thus, the LCM of 22, 54, 108, 135, and 198 is **5940**.

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