To find the LCM (Least Common Multiple) of the numbers 22, 54, 135, and 198, we can use the prime factorization method. Here’s a step-by-step solution:
### Step 1: Prime Factorization
We need to find the prime factorization of each number.
- **22**:
- 22 = 2 × 11
- **54**:
- 54 = 2 × 3^3 (since 54 = 2 × 27 and 27 = 3 × 3 × 3)
- **135**:
- 135 = 3^3 × 5 (since 135 = 27 × 5 and 27 = 3 × 3 × 3)
- **198**:
- 198 = 2 × 3^2 × 11 (since 198 = 2 × 99 and 99 = 9 × 11, where 9 = 3 × 3)
### Step 2: List the Prime Factors
Now, we list all the prime factors with their highest powers from each number:
- From 22: 2^1, 11^1
- From 54: 2^1, 3^3
- From 135: 3^3, 5^1
- From 198: 2^1, 3^2, 11^1
### Step 3: Determine the Highest Power of Each Prime Factor
Now we take the highest power of each prime factor:
- **2**: The highest power is 2^1
- **3**: The highest power is 3^3
- **5**: The highest power is 5^1
- **11**: The highest power is 11^1
### Step 4: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:
LCM = 2^1 × 3^3 × 5^1 × 11^1
Calculating this step-by-step:
- 2^1 = 2
- 3^3 = 27
- 5^1 = 5
- 11^1 = 11
Now multiply these together:
1. First, multiply 2 and 27:
- 2 × 27 = 54
2. Next, multiply the result by 5:
- 54 × 5 = 270
3. Finally, multiply the result by 11:
- 270 × 11 = 2970
So, the LCM of 22, 54, 135, and 198 is **2970**.
### Final Answer:
**LCM = 2970**
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